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Mathematics · Textbook solutions

Determinants and Matrices

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. · 264 questions

4.1.1 Value of a Determinant

4 q

Solved Examples

Worked · 4
  1. Evaluate
    4.1.1 SolvedEx.1(i)
    ∣79−43∣\begin{vmatrix}7 & 9\\-4 & 3\end{vmatrix}
  2. 4.1.1 SolvedEx.1(ii)
    ∣cos⁡θsin⁡θ−sin⁡θcos⁡θ∣\begin{vmatrix}\cos\theta & \sin\theta\\-\sin\theta & \cos\theta\end{vmatrix}
  3. 4.1.1 SolvedEx.1(iii)
    ∣4i−2i7∣\begin{vmatrix}4 & i\\-2i & 7\end{vmatrix} where i2=−1i^2 = -1
  4. 4.1.1 SolvedEx.1(iv)
    ∣log⁡42log⁡4224∣\begin{vmatrix}\log_4 2 & \log_4 2\\2 & 4\end{vmatrix}

4.1.2 Determinant of Order Three

3 q

Solved Examples

Worked · 3
  1. Evaluate:
    4.1.2 SolvedEx.1(i)
    ∣3−4511−223−1∣\begin{vmatrix}3 & -4 & 5\\1 & 1 & -2\\2 & 3 & -1\end{vmatrix}
  2. 4.1.2 SolvedEx.1(ii)
    ∣sec⁡θtan⁡θ0tan⁡θsec⁡θ0001∣\begin{vmatrix}\sec\theta & \tan\theta & 0\\\tan\theta & \sec\theta & 0\\0 & 0 & 1\end{vmatrix}
  3. 4.1.2 SolvedEx.1(iii)
    ∣2−i3−132−i02−12−i∣\begin{vmatrix}2-i & 3 & -1\\3 & 2-i & 0\\2 & -1 & 2-i\end{vmatrix} where i=−1i = \sqrt{-1}

4.1.3 Minors and Cofactors

5 q

Solved Examples

Worked · 5
  1. Find Minors and Cofactors of the elements of determinant
    4.1.3 SolvedEx.1(i)
    ∣2−347∣\begin{vmatrix}2 & -3\\4 & 7\end{vmatrix}
  2. 4.1.3 SolvedEx.1(ii)
    ∣12−3−2045−13∣\begin{vmatrix}1 & 2 & -3\\-2 & 0 & 4\\5 & -1 & 3\end{vmatrix}
  3. Find value of x if
    4.1.3 SolvedEx.2(i)
    ∣x−122x1−33−45∣=−10\begin{vmatrix}x & -1 & 2\\2x & 1 & -3\\3 & -4 & 5\end{vmatrix} = -10
  4. 4.1.3 SolvedEx.2(ii)
    ∣x32xx1101∣=9\begin{vmatrix}x & 3 & 2\\x & x & 1\\1 & 0 & 1\end{vmatrix} = 9
  5. 4.1.3 SolvedEx.3
    Find the value of ∣1−12−235−20−1∣\begin{vmatrix}1 & -1 & 2\\-2 & 3 & 5\\-2 & 0 & -1\end{vmatrix} by expanding along a) 2nd2^{\text{nd}} row b) 3rd3^{\text{rd}} column and Interprete the result.

Exercise 4.1

10 q
  1. Find the value of determinant
    Ex 4.1 Q1(i)
    ∣2−47−15∣\begin{vmatrix}2 & -4\\7 & -15\end{vmatrix}
  2. Ex 4.1 Q1(ii)
    ∣2i34−i∣\begin{vmatrix}2i & 3\\4 & -i\end{vmatrix}
  3. Ex 4.1 Q1(iii)
    ∣3−4511−2231∣\begin{vmatrix}3 & -4 & 5\\1 & 1 & -2\\2 & 3 & 1\end{vmatrix}
  4. Ex 4.1 Q1(iv)
    ∣ahghbfgfc∣\begin{vmatrix}a & h & g\\h & b & f\\g & f & c\end{vmatrix}
  5. Find the value of xx if
    Ex 4.1 Q2(i)
    ∣x2−x+1x+1x+1x+1∣=0\begin{vmatrix}x^2 - x + 1 & x + 1\\x + 1 & x + 1\end{vmatrix} = 0
  6. Ex 4.1 Q2(ii)
    ∣x−122x1−33−45∣=29\begin{vmatrix}x & -1 & 2\\2x & 1 & -3\\3 & -4 & 5\end{vmatrix} = 29
  7. Ex 4.1 Q3
    Find xx and yy if ∣4ii32i13i245−3i∣=x+iy\begin{vmatrix}4i & i^3 & 2i\\1 & 3i^2 & 4\\5 & -3 & i\end{vmatrix} = x + iy where i2=−1i^2 = -1
  8. Ex 4.1 Q4
    Find the minor and cofactor of element of the determinant D=∣2−1312−1572∣D = \begin{vmatrix}2 & -1 & 3\\1 & 2 & -1\\5 & 7 & 2\end{vmatrix}
  9. Ex 4.1 Q5
    Evaluate A=∣2−3560415−7∣A = \begin{vmatrix}2 & -3 & 5\\6 & 0 & 4\\1 & 5 & -7\end{vmatrix} Also find minor and cofactor of elements in the 2nd2^{\text{nd}} row of determinant and verify a) −a21⋅M21+a22⋅M22−a23⋅M23=-a_{21} \cdot M_{21} + a_{22} \cdot M_{22} - a_{23} \cdot M_{23} = value of A b) a21C21+a22C22+a23C23=a_{21} C_{21} + a_{22} C_{22} + a_{23} C_{23} = value of A where M21, M22, M23M_{21},\ M_{22},\ M_{23} are minor of a21, a22, a23a_{21},\ a_{22},\ a_{23} and C21, C22, C23C_{21},\ C_{22},\ C_{23} are cofactor of a21, a22, a23a_{21},\ a_{22},\ a_{23}
  10. Ex 4.1 Q6
    Find the value of determinant expanding along third column ∣−112−23−4−340∣\begin{vmatrix}-1 & 1 & 2\\-2 & 3 & -4\\-3 & 4 & 0\end{vmatrix}

4.2 Properties of Determinants

4 q

Solved Examples

Worked · 4
  1. Show that
    4.2 SolvedEx.1(i)
    ∣101202303505606707123∣=0\begin{vmatrix}101 & 202 & 303\\505 & 606 & 707\\1 & 2 & 3\end{vmatrix} = 0
  2. 4.2 SolvedEx.1(ii)
    ∣312313314315316317318319320∣=0\begin{vmatrix}312 & 313 & 314\\315 & 316 & 317\\318 & 319 & 320\end{vmatrix} = 0
  3. 4.2 SolvedEx.2
    Prove that ∣1abc1bca1cab∣=∣1aa21bb21cc2∣\begin{vmatrix}1 & a & bc\\1 & b & ca\\1 & c & ab\end{vmatrix} = \begin{vmatrix}1 & a & a^2\\1 & b & b^2\\1 & c & c^2\end{vmatrix}
  4. 4.2 SolvedEx.3
    If ∣xyz−xyzx−yz∣=k⋅xyz\begin{vmatrix}x & y & z\\-x & y & z\\x & -y & z\end{vmatrix} = k \cdot xyz then find the value of kk

Exercise 4.2

10 q
  1. Without expanding evaluate the following determinants.
    Ex 4.2 Q1(i)
    ∣1ab+c1bc+a1ca+b∣\begin{vmatrix}1 & a & b+c\\ 1 & b & c+a\\ 1 & c & a+b\end{vmatrix}
  2. Ex 4.2 Q1(ii)
    ∣2345686x9x12x∣\begin{vmatrix}2 & 3 & 4\\ 5 & 6 & 8\\ 6x & 9x & 12x\end{vmatrix}
  3. Ex 4.2 Q1(iii)
    ∣276538755986∣\begin{vmatrix}2 & 7 & 65\\ 3 & 8 & 75\\ 5 & 9 & 86\end{vmatrix}
  4. Ex 4.2 Q2
    Prove that ∣x+yy+zz+xz+xx+yy+zy+zz+xx+y∣=2∣xyzzxyyzx∣\begin{vmatrix}x+y & y+z & z+x\\ z+x & x+y & y+z\\ y+z & z+x & x+y\end{vmatrix} = 2\begin{vmatrix}x & y & z\\ z & x & y\\ y & z & x\end{vmatrix}
  5. Using properties of determinant show that
    Ex 4.2 Q3(i)
    ∣a+babaa+ccbcb+c∣=4abc\begin{vmatrix}a+b & a & b\\ a & a+c & c\\ b & c & b+c\end{vmatrix} = 4abc
  6. Ex 4.2 Q3(ii)
    ∣1log⁡xylog⁡xzlog⁡yx1log⁡yzlog⁡zxlog⁡zy1∣=0\begin{vmatrix}1 & \log_x y & \log_x z\\ \log_y x & 1 & \log_y z\\ \log_z x & \log_z y & 1\end{vmatrix} = 0
  7. Solve the following equations.
    Ex 4.2 Q5(i)
    ∣x+2x+6x−1x+6x−1x+2x−1x+2x+6∣=0\begin{vmatrix}x+2 & x+6 & x-1\\ x+6 & x-1 & x+2\\ x-1 & x+2 & x+6\end{vmatrix} = 0
  8. Ex 4.2 Q5(ii)
    ∣x−1xx−20x−2x−300x−3∣=0\begin{vmatrix}x-1 & x & x-2\\ 0 & x-2 & x-3\\ 0 & 0 & x-3\end{vmatrix} = 0
  9. Ex 4.2 Q6
    If ∣4+x4−x4−x4−x4+x4−x4−x4−x4+x∣=0\begin{vmatrix}4+x & 4-x & 4-x\\ 4-x & 4+x & 4-x\\ 4-x & 4-x & 4+x\end{vmatrix} = 0 then find the values of xx.
  10. Ex 4.2 Q7
    Without expanding determinants show that ∣1366143712∣+4∣233212176∣=10∣121317326∣\begin{vmatrix}1 & 3 & 6\\ 6 & 1 & 4\\ 3 & 7 & 12\end{vmatrix} + 4\begin{vmatrix}2 & 3 & 3\\ 2 & 1 & 2\\ 1 & 7 & 6\end{vmatrix} = 10\begin{vmatrix}1 & 2 & 1\\ 3 & 1 & 7\\ 3 & 2 & 6\end{vmatrix}

4.3.1 Cramer's Rule

4 q

Solved Examples

Worked · 4
  1. 4.3.1 SolvedEx.1
    Solve the following equation by using Cramer's rule. x+y+z=6,x−y+z=2,x+2y−z=2x+y+z = 6,\quad x-y+z = 2,\quad x+2y-z = 2
  2. 4.3.1 SolvedEx.2
    By using Cramer's rule solve the following linear equations. x+y−z=1,8x+3y−6z=1,−4x−y+3z=1x+y-z = 1,\quad 8x+3y-6z = 1,\quad -4x-y+3z = 1
  3. 4.3.1 SolvedEx.3
    Solve the following equations by using determinant 1x+1y+1z=−2,1x−2y+1z=3,2x−1y+32=−1\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} = -2,\quad \dfrac{1}{x}-\dfrac{2}{y}+\dfrac{1}{z} = 3,\quad \dfrac{2}{x}-\dfrac{1}{y}+\dfrac{3}{2} = -1
  4. 4.3.1 SolvedEx.4
    The cost of 2 books, 6 notebooks and 3 pens is Rs.120. The cost of 3 books, 4 notebooks and 2 pens is Rs.105, while the cost of 5 books, 7 notebooks and 4 pens is Rs.183. Using this information find the cost of 1 book, 1 notebook and 1 pen.

4.3.2 Consistency of Three Equations

5 q

Solved Examples

Worked · 5
  1. 4.3.2 SolvedEx.1
    Verify the consistency of following equations 2x+2y=−2,x+y=−1,3x+3y=−52x+2y = -2,\quad x + y = -1,\quad 3x + 3y = -5
  2. Examine the consistency of following equations.
    4.3.2 SolvedEx.2(i)
    x+y=2,2x+3y=5,3x−2y=1x + y = 2,\quad 2x + 3y = 5,\quad 3x - 2y = 1
  3. 4.3.2 SolvedEx.2(ii)
    x+2y−3=0,7x+4y−11=0,2x+3y+1=0x + 2y - 3 = 0,\quad 7x + 4y - 11 = 0,\quad 2x + 3y + 1 = 0
  4. 4.3.2 SolvedEx.2(iii)
    x+y=1,2x+2y=2,3x+3y=5x + y = 1,\quad 2x + 2y = 2,\quad 3x + 3y = 5
  5. 4.3.2 SolvedEx.3
    Find the value of k if the following equations are consistent. 7x−ky=4,2x+5y=97x - ky = 4,\quad 2x + 5y = 9 and 2x+y=82x + y = 8

4.3.3 Area of a Triangle and Collinearity

4 q

Solved Examples

Worked · 4
  1. 4.3.3 SolvedEx.1
    Find the area of the triangle whose vertices are A(−2,−3)(-2, -3), B(3,2)(3, 2) and C(−1,−8)(-1, -8).
  2. 4.3.3 SolvedEx.2
    If the area of triangle with vertices P(−3,0)(-3, 0), Q(3,0)(3, 0) and R(0,K)(0, K) is 9 square unit then find the value of k.
  3. 4.3.3 SolvedEx.3
    Find the area of triangle whose vertices are A(3,7)(3, 7), B(4,−3)(4, -3) and C(5,−13)(5, -13). Interpret your answer.
  4. 4.3.3 SolvedEx.4
    Show that the following points are collinear by determinant method. A(2,5)(2, 5), B(5,7)(5, 7), C(8,9)(8, 9)

Exercise 4.3

18 q
  1. Solve the following linear equations by using Cramer's Rule.
    Ex 4.3 Q1(i)
    x+y+z=6,  x−y+z=2,  x+2y−z=2x+y+z = 6,\; x-y+z = 2,\; x+2y-z = 2
  2. Ex 4.3 Q1(ii)
    x+y−2z=−10,  2x+y−3z=−19,  4x+6y+z=2x+y-2z = -10,\; 2x+y-3z = -19,\; 4x+6y+z = 2
  3. Ex 4.3 Q1(iii)
    x+z=1,  y+z=1,  x+y=4x+z = 1,\; y+z = 1,\; x+y = 4
  4. Ex 4.3 Q1(iv)
    −2x−1y−3z=3,2x−3y+1z=−13\dfrac{-2}{x}-\dfrac{1}{y}-\dfrac{3}{z} = 3,\quad \dfrac{2}{x}-\dfrac{3}{y}+\dfrac{1}{z} = -13 and 2x−3z=−11\dfrac{2}{x}-\dfrac{3}{z} = -11
  5. Ex 4.3 Q2
    The sum of three numbers is 15. If the second number is subtracted from the sum of first and third numbers then we get 5. When the third number is subtracted from the sum of twice the first number and the second number, we get 4. Find the three numbers.
  6. Examine the consistency of the following equations.
    Ex 4.3 Q3(i)
    2x−y+3=0,  3x+y−2=0,  11x+2y−3=02x-y+3 = 0,\; 3x+y-2 = 0,\; 11x+2y-3 = 0
  7. Ex 4.3 Q3(ii)
    2x+3y−4=0,  x+2y=3,  3x+4y+5=02x+3y-4 = 0,\; x+2y = 3,\; 3x+4y+5 = 0
  8. Ex 4.3 Q3(iii)
    x+2y−3=0,  7x+4y−11=0,  2x+4y−6=0x+2y-3 = 0,\; 7x+4y-11 = 0,\; 2x+4y-6 = 0
  9. Find k if the following equations are consistent.
    Ex 4.3 Q4(i)
    2x+3y−2=0,  2x+4y−k=0,  x−2y+3k=02x+3y-2 = 0,\; 2x+4y-k = 0,\; x-2y+3k = 0
  10. Ex 4.3 Q4(ii)
    kx+3y+1=0,  x+2y+1=0,  x+y=0kx+3y+1 = 0,\; x+2y+1 = 0,\; x+y = 0
  11. Find the area of triangle whose vertices are
    Ex 4.3 Q5(i)
    A(5,8)(5, 8), B(5,0)(5, 0), C(1,0)(1, 0)
  12. Ex 4.3 Q5(ii)
    P(32,1)\left(\dfrac{3}{2}, 1\right), Q(4,2)(4, 2), R(4,−12)\left(4, \dfrac{-1}{2}\right)
  13. Ex 4.3 Q5(iii)
    M(0,5)(0, 5), N(−2,3)(-2, 3), T(1,−4)(1, -4)
  14. Ex 4.3 Q6
    Find the area of quadrilateral whose vertices are A(−3,1)(-3, 1), B(−2,−2)(-2, -2), C(3,−1)(3, -1), D(1,4)(1, 4).
  15. Ex 4.3 Q7
    Find the value of k, if the area of triangle whose vertices are P(k,0)(k, 0), Q(2,2)(2, 2), R(4,3)(4, 3) is 32\dfrac{3}{2} sq.unit.
  16. Examine the collinearity of the following set of points
    Ex 4.3 Q8(i)
    A(3,−1)(3, -1), B(0,−3)(0, -3), C(12,5)(12, 5)
  17. Ex 4.3 Q8(ii)
    P(3,−5)(3, -5), Q(6,1)(6, 1), R(4,2)(4, 2)
  18. Ex 4.3 Q8(iii)
    L(0,12)\left(0, \dfrac{1}{2}\right), M(2,−1)(2, -1), N(−4,72)\left(-4, \dfrac{7}{2}\right)

Miscellaneous Exercise 4 (A)

42 q

(I) Select the correct option

Practice · 10
  1. Misc 4A I Q1
    The determinant D=∣aba+bbcb+ca+bb+c0∣=0D = \begin{vmatrix} a & b & a+b \\ b & c & b+c \\ a+b & b+c & 0 \end{vmatrix} = 0 if

    Tap an option to check your answer.

  2. Misc 4A I Q2
    If ∣xkxk+2xk+3ykyk+2yk+3zkzk+2zk+3∣=(x−y)(y−z)(z−x)(1x+1y+1z)\begin{vmatrix} x^{k} & x^{k+2} & x^{k+3} \\ y^{k} & y^{k+2} & y^{k+3} \\ z^{k} & z^{k+2} & z^{k+3} \end{vmatrix} = (x-y)(y-z)(z-x)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) then

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  3. Misc 4A I Q3
    Let D=∣sin⁡θ⋅cos⁡ϕsin⁡θ⋅sin⁡ϕcos⁡θcos⁡θ⋅cos⁡ϕcos⁡θ⋅sin⁡ϕ−sin⁡θ−sin⁡θ⋅sin⁡ϕsin⁡θ⋅cos⁡ϕ0∣D = \begin{vmatrix} \sin\theta\cdot\cos\phi & \sin\theta\cdot\sin\phi & \cos\theta \\ \cos\theta\cdot\cos\phi & \cos\theta\cdot\sin\phi & -\sin\theta \\ -\sin\theta\cdot\sin\phi & \sin\theta\cdot\cos\phi & 0 \end{vmatrix} then

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  4. Misc 4A I Q4
    The value of aa for which system of equation a3x+(a+1)3y+(a+2)3z=0a^3x + (a+1)^3y + (a+2)^3z = 0, ax+(a+1)y+(a+2)z=0ax + (a+1)y + (a+2)z = 0 and x+y+z=0x + y + z = 0 has non zero Soln. is

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  5. Misc 4A I Q5
    ∣b+cc+aa+bq+rr+pp+qy+zz+xx+y∣=\begin{vmatrix} b+c & c+a & a+b \\ q+r & r+p & p+q \\ y+z & z+x & x+y \end{vmatrix} =

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  6. Misc 4A I Q6
    The system 3x−y+4z=33x - y + 4z = 3, x+2y−3z=−2x + 2y - 3z = -2 and 6x+5y+λz=−36x + 5y + \lambda z = -3 has at least one Solution when

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  7. Misc 4A I Q7
    If x=−9x = -9 is a root of ∣x372x276x∣=0\begin{vmatrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{vmatrix} = 0 then other two roots are

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  8. Misc 4A I Q8
    If ∣6i−3i143i−1203i∣=x+iy\begin{vmatrix} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \end{vmatrix} = x + iy then

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  9. Misc 4A I Q9
    If A(0,0)A(0,0), B(1,3)B(1,3) and C(k,0)C(k,0) are vertices of triangle ABC whose area is 3 sq.units then value of kk is

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  10. Misc 4A I Q10
    Which of the following is correct

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(II) Answer the following questions

Practice · 32
  1. Evaluate.
    Misc 4A II Q1(i)
    ∣2−57521902∣\begin{vmatrix} 2 & -5 & 7 \\ 5 & 2 & 1 \\ 9 & 0 & 2 \end{vmatrix}
  2. Misc 4A II Q1(ii)
    ∣1−31202−4972∣\begin{vmatrix} 1 & -3 & 12 \\ 0 & 2 & -4 \\ 9 & 7 & 2 \end{vmatrix}
  3. Misc 4A II Q2
    Evaluate determinant along second column ∣1−1232−201−2∣\begin{vmatrix} 1 & -1 & 2 \\ 3 & 2 & -2 \\ 0 & 1 & -2 \end{vmatrix}
  4. Evaluate by using properties.
    Misc 4A II Q3(i)
    ∣2354006001000484718∣\begin{vmatrix} 2 & 3 & 5 \\ 400 & 600 & 1000 \\ 48 & 47 & 18 \end{vmatrix}
  5. Misc 4A II Q3(ii)
    ∣101102103106107108123∣\begin{vmatrix} 101 & 102 & 103 \\ 106 & 107 & 108 \\ 1 & 2 & 3 \end{vmatrix}
  6. Find minor and cofactor of elements of the determinant.
    Misc 4A II Q4(i)
    ∣−104−2130−42∣\begin{vmatrix} -1 & 0 & 4 \\ -2 & 1 & 3 \\ 0 & -4 & 2 \end{vmatrix}
  7. Misc 4A II Q4(ii)
    ∣1−1230−2103∣\begin{vmatrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{vmatrix}
  8. Find the value of xx if
    Misc 4A II Q5(i)
    ∣14201−2−512x5x2∣=0\begin{vmatrix} 1 & 4 & 20 \\ 1 & -2 & -5 \\ 1 & 2x & 5x^{2} \end{vmatrix} = 0
  9. Misc 4A II Q5(ii)
    ∣12x4x1416111∣=0\begin{vmatrix} 1 & 2x & 4x \\ 1 & 4 & 16 \\ 1 & 1 & 1 \end{vmatrix} = 0
  10. Misc 4A II Q6
    By using properties of determinant prove that ∣x+yy+zz+xzxy111∣=0\begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix} = 0
  11. Without expanding determinant show that
    Misc 4A II Q7(i)
    ∣b+cbcb2c2c+acac2a2a+baba2b2∣=0\begin{vmatrix} b+c & bc & b^{2}c^{2} \\ c+a & ca & c^{2}a^{2} \\ a+b & ab & a^{2}b^{2} \end{vmatrix} = 0
  12. Misc 4A II Q7(ii)
    ∣xaybzca2b2c2111∣=∣xyzabcbccaab∣\begin{vmatrix} xa & yb & zc \\ a^{2} & b^{2} & c^{2} \\ 1 & 1 & 1 \end{vmatrix} = \begin{vmatrix} x & y & z \\ a & b & c \\ bc & ca & ab \end{vmatrix}
  13. Misc 4A II Q7(iii)
    ∣lmnedfuvw∣=∣nfwleumdv∣\begin{vmatrix} l & m & n \\ e & d & f \\ u & v & w \end{vmatrix} = \begin{vmatrix} n & f & w \\ l & e & u \\ m & d & v \end{vmatrix}
  14. Misc 4A II Q7(iv)
    ∣0ab−a0c−b−c0∣=0\begin{vmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{vmatrix} = 0
  15. Misc 4A II Q8
    If ∣a111b111c∣=0\begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{vmatrix} = 0 then show that 11−a+11−b+11−c=1\frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c} = 1
  16. Solve the following linear equations by Cramer's Rule.
    Misc 4A II Q9(i)
    2x−y+z=12x - y + z = 1, x+2y+3z=8x + 2y + 3z = 8, 3x+y−4z=13x + y - 4z = 1
  17. Misc 4A II Q9(ii)
    1x+1y=32\frac{1}{x} + \frac{1}{y} = \frac{3}{2}, 1y+1z=56\frac{1}{y} + \frac{1}{z} = \frac{5}{6}, 1z+1x=43\frac{1}{z} + \frac{1}{x} = \frac{4}{3}
  18. Misc 4A II Q9(iii)
    2x+3y+3z=52x + 3y + 3z = 5, x−2y+z=−4x - 2y + z = -4, 3x−y−2z=33x - y - 2z = 3
  19. Misc 4A II Q9(iv)
    x−y+2z=7x - y + 2z = 7, 3x+4y−5z=53x + 4y - 5z = 5, 2x−y+3z=122x - y + 3z = 12
  20. Find the value of kk if the following equations are consistent.
    Misc 4A II Q10(i)
    (k+1)x+(k−1)y+(k−1)=0(k+1)x + (k-1)y + (k-1) = 0, (k−1)x+(k+1)y+(k−1)=0(k-1)x + (k+1)y + (k-1) = 0, (k−1)x+(k−1)y+(k+1)=0(k-1)x + (k-1)y + (k+1) = 0
  21. Misc 4A II Q10(ii)
    3x+y−2=03x + y - 2 = 0, kx+2y−3=0kx + 2y - 3 = 0 and 2x−y=32x - y = 3
  22. Misc 4A II Q10(iii)
    (k−2)x+(k−1)y=17(k-2)x + (k-1)y = 17, (k−1)x+(k−2)y=18(k-1)x + (k-2)y = 18 and x+y=5x + y = 5
  23. Find the area of triangle whose vertices are
    Misc 4A II Q11(i)
    A(−1,2)A(-1, 2), B(2,4)B(2, 4), C(0,0)C(0, 0)
  24. Misc 4A II Q11(ii)
    P(3,6)P(3, 6), Q(−1,3)Q(-1, 3), R(2,−1)R(2, -1)
  25. Misc 4A II Q11(iii)
    L(1,1)L(1, 1), M(−2,2)M(-2, 2), N(5,4)N(5, 4)
  26. Find the value of kk
    Misc 4A II Q12(i)
    If area of triangle is 4 square unit and vertices are P(k,0)P(k, 0), Q(4,0)Q(4, 0), R(0,2)R(0, 2)
  27. Misc 4A II Q12(ii)
    If area of triangle is 332\frac{33}{2} square unit and vertices are L(3,−5)L(3, -5), M(−2,k)M(-2, k), N(1,4)N(1, 4)
  28. Misc 4A II Q13
    Find the area of quadrilateral whose vertices are A(0,−4)A(0, -4), B(4,0)B(4, 0), C(−4,0)C(-4, 0), D(0,4)D(0, 4)
  29. Misc 4A II Q14
    An amount of ₹ 5000 is put into three investments at the rate of interest of 6%, 7% and 8% per annum respectively. The total annual income is ₹ 350. If the combined income from the first two investments is ₹ 70 more than the income from the third. Find the amount of each investment.
  30. Misc 4A II Q15
    Show that the lines x−y=6x - y = 6, 4x−3y=204x - 3y = 20 and 6x+5y+8=06x + 5y + 8 = 0 are concurrent. Also find the point of concurrence
  31. Show that the following points are collinear by determinant
    Misc 4A II Q16(a)
    L(2,5)L(2, 5), M(5,7)M(5, 7), N(8,9)N(8, 9)
  32. Misc 4A II Q16(b)
    P(5,1)P(5, 1), Q(1,−1)Q(1, -1), R(11,4)R(11, 4)

4.4 Introduction to Matrices

3 q

Solved Examples

Worked · 3
  1. 4.4 SolvedEx.1
    Show that the matrix [x+yy+zz+x111zxy]\begin{bmatrix} x+y & y+z & z+x \\ 1 & 1 & 1 \\ z & x & y \end{bmatrix} is a singular matrix.
  2. 4.4 SolvedEx.2
    If A=[−1−5203−4]3×2A = \begin{bmatrix} -1 & -5 \\ 2 & 0 \\ 3 & -4 \end{bmatrix}_{3\times 2}, find (AT)T(A^{T})^{T}.
  3. 4.4 SolvedEx.3
    Find aa, bb, cc if the matrix A=[2a3−745cb6]A = \begin{bmatrix} 2 & a & 3 \\ -7 & 4 & 5 \\ c & b & 6 \end{bmatrix} is a symmetric matrix.

Exercise 4.4

28 q
  1. Construct a matrix A=[aij]3×2A = [a_{ij}]_{3\times 2} whose elements aija_{ij} are given by
    Ex 4.4 Q1(i)
    aij=(i−j)25−ia_{ij} = \frac{(i-j)^2}{5-i}
  2. Ex 4.4 Q1(ii)
    aij=i−3ja_{ij} = i - 3j
  3. Ex 4.4 Q1(iii)
    aij=(i+j)35a_{ij} = \frac{(i+j)^3}{5}
  4. Classify the following matrices as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix.
    Ex 4.4 Q2(i)
    [3−2400−5000]\begin{bmatrix} 3 & -2 & 4 \\ 0 & 0 & -5 \\ 0 & 0 & 0 \end{bmatrix}
  5. Ex 4.4 Q2(ii)
    [047−40−3−730]\begin{bmatrix} 0 & 4 & 7 \\ -4 & 0 & -3 \\ -7 & 3 & 0 \end{bmatrix}
  6. Ex 4.4 Q2(iii)
    [54−3]\begin{bmatrix} 5 \\ 4 \\ -3 \end{bmatrix}
  7. Ex 4.4 Q2(iv)
    [92−3]\begin{bmatrix} 9 & \sqrt{2} & -3 \end{bmatrix}
  8. Ex 4.4 Q2(v)
    [6006]\begin{bmatrix} 6 & 0 \\ 0 & 6 \end{bmatrix}
  9. Ex 4.4 Q2(vi)
    [2003−10−731]\begin{bmatrix} 2 & 0 & 0 \\ 3 & -1 & 0 \\ -7 & 3 & 1 \end{bmatrix}
  10. Ex 4.4 Q2(vii)
    [3000500013]\begin{bmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & \frac{1}{3} \end{bmatrix}
  11. Ex 4.4 Q2(viii)
    [10−1527−15034273453]\begin{bmatrix} 10 & -15 & 27 \\ -15 & 0 & \sqrt{34} \\ 27 & \sqrt{34} & \frac{5}{3} \end{bmatrix}
  12. Ex 4.4 Q2(ix)
    [100010001]\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}
  13. Ex 4.4 Q2(x)
    [001010100]\begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}
  14. Which of the following matrices are singular or non singular ?
    Ex 4.4 Q3(i)
    [abcpqr2a−p2b−q2c−r]\begin{bmatrix} a & b & c \\ p & q & r \\ 2a-p & 2b-q & 2c-r \end{bmatrix}
  15. Ex 4.4 Q3(ii)
    [505199100699105]\begin{bmatrix} 5 & 0 & 5 \\ 1 & 99 & 100 \\ 6 & 99 & 105 \end{bmatrix}
  16. Ex 4.4 Q3(iii)
    [357−214325]\begin{bmatrix} 3 & 5 & 7 \\ -2 & 1 & 4 \\ 3 & 2 & 5 \end{bmatrix}
  17. Ex 4.4 Q3(iv)
    [75−47]\begin{bmatrix} 7 & 5 \\ -4 & 7 \end{bmatrix}
  18. Find kk if the following matrices are singular
    Ex 4.4 Q4(i)
    [73−2k]\begin{bmatrix} 7 & 3 \\ -2 & k \end{bmatrix}
  19. Ex 4.4 Q4(ii)
    [4317k11091]\begin{bmatrix} 4 & 3 & 1 \\ 7 & k & 1 \\ 10 & 9 & 1 \end{bmatrix}
  20. Ex 4.4 Q4(iii)
    [k−1233121−24]\begin{bmatrix} k-1 & 2 & 3 \\ 3 & 1 & 2 \\ 1 & -2 & 4 \end{bmatrix}
  21. Ex 4.4 Q5
    If A=[51−1320]A = \begin{bmatrix} 5 & 1 & -1 \\ 3 & 2 & 0 \end{bmatrix}, find (AT)T(A^{T})^{T}.
  22. Ex 4.4 Q6
    If A=[731−2−41591]A = \begin{bmatrix} 7 & 3 & 1 \\ -2 & -4 & 1 \\ 5 & 9 & 1 \end{bmatrix}, find (AT)T(A^{T})^{T}.
  23. Ex 4.4 Q7
    Find aa, bb, cc if [135ab−5−7−4c0]\begin{bmatrix} 1 & \frac{3}{5} & a \\ b & -5 & -7 \\ -4 & c & 0 \end{bmatrix} is a symmetric matrix.
  24. Ex 4.4 Q8
    Find xx, yy, zz if [0−5ixy0z32−20]\begin{bmatrix} 0 & -5i & x \\ y & 0 & z \\ \frac{3}{2} & -\sqrt{2} & 0 \end{bmatrix} is a skew symmetric matrix.
  25. For each of the following matrices, using its transpose state whether it is a symmetric, a skew-symmetric or neither.
    Ex 4.4 Q9(i)
    [12−52−34−549]\begin{bmatrix} 1 & 2 & -5 \\ 2 & -3 & 4 \\ -5 & 4 & 9 \end{bmatrix}
  26. Ex 4.4 Q9(ii)
    [251−546−1−63]\begin{bmatrix} 2 & 5 & 1 \\ -5 & 4 & 6 \\ -1 & -6 & 3 \end{bmatrix}
  27. Ex 4.4 Q9(iii)
    [01+2ii−2−1−2i0−72−i70]\begin{bmatrix} 0 & 1+2i & i-2 \\ -1-2i & 0 & -7 \\ 2-i & 7 & 0 \end{bmatrix}
  28. Ex 4.4 Q10
    Construct the matrix A=[aij]3×3A = [a_{ij}]_{3\times 3} where aij=i−ja_{ij} = i-j. State whether A is symmetric or skew symmetric.

4.5 Algebra of Matrices

7 q

Worked Examples

Worked · 2
  1. 4.5.3 SolvedEx.1
    A=[231−1−20]2×3A = \begin{bmatrix} 2 & 3 & 1 \\ -1 & -2 & 0 \end{bmatrix}_{2\times 3} and B=[−43157−8]2×3B = \begin{bmatrix} -4 & 3 & 1 \\ 5 & 7 & -8 \end{bmatrix}_{2\times 3}. Find A+BA+B.
  2. 4.5.3 SolvedEx.2
    If A=[−143−205]3×2A = \begin{bmatrix} -1 & 4 \\ 3 & -2 \\ 0 & 5 \end{bmatrix}_{3\times 2} and B=[−152−649]3×2B = \begin{bmatrix} -1 & 5 \\ 2 & -6 \\ 4 & 9 \end{bmatrix}_{3\times 2}, find A−BA-B.

Solved Examples

Worked · 5
  1. 4.5 SolvedEx.1
    If A=[5−310−4−2]A = \begin{bmatrix} 5 & -3 \\ 1 & 0 \\ -4 & -2 \end{bmatrix} and B=[27−312−2]B = \begin{bmatrix} 2 & 7 \\ -3 & 1 \\ 2 & -2 \end{bmatrix}, find 2A−3B2A-3B.
  2. 4.5 SolvedEx.2
    If A=diag⁡(2,−5,9)A = \operatorname{diag}(2, -5, 9), B=diag⁡(−3,7,−14)B = \operatorname{diag}(-3, 7, -14) and C=diag⁡(1,0,3)C = \operatorname{diag}(1, 0, 3), find B−A−CB-A-C.
  3. 4.5 SolvedEx.3
    If A=[23−1475]A = \begin{bmatrix} 2 & 3 & -1 \\ 4 & 7 & 5 \end{bmatrix}, B=[13246−1]B = \begin{bmatrix} 1 & 3 & 2 \\ 4 & 6 & -1 \end{bmatrix} and C=[1−1602−5]C = \begin{bmatrix} 1 & -1 & 6 \\ 0 & 2 & -5 \end{bmatrix}, find the matrix XX such that 3A−2B+4X=5C3A-2B+4X = 5C.
  4. 4.5 SolvedEx.4
    If [2x+1−134y]+[−1630]=[45612]\begin{bmatrix} 2x+1 & -1 \\ 3 & 4y \end{bmatrix} + \begin{bmatrix} -1 & 6 \\ 3 & 0 \end{bmatrix} = \begin{bmatrix} 4 & 5 \\ 6 & 12 \end{bmatrix}, find xx and yy.
  5. 4.5 SolvedEx.5
    If X+Y=[2−113−3−2]X+Y = \begin{bmatrix} 2 & -1 \\ 1 & 3 \\ -3 & -2 \end{bmatrix} and X−2Y=[−213−14−2]X-2Y = \begin{bmatrix} -2 & 1 \\ 3 & -1 \\ 4 & -2 \end{bmatrix}, then find XX, YY.

Exercise 4.5

13 q
  1. If A=[2−35−4−61]A = \begin{bmatrix} 2 & -3 \\ 5 & -4 \\ -6 & 1 \end{bmatrix}, B=[−122203]B = \begin{bmatrix} -1 & 2 \\ 2 & 2 \\ 0 & 3 \end{bmatrix} and C=[43−14−21]C = \begin{bmatrix} 4 & 3 \\ -1 & 4 \\ -2 & 1 \end{bmatrix}, show that
    Ex 4.5 Q1(i)
    A+B=B+AA + B = B + A
  2. Ex 4.5 Q1(ii)
    (A+B)+C=A+(B+C)(A + B) + C = A + (B + C)
  3. Ex 4.5 Q2
    If A=[1−253]A = \begin{bmatrix} 1 & -2 \\ 5 & 3 \end{bmatrix}, B=[1−34−7]B = \begin{bmatrix} 1 & -3 \\ 4 & -7 \end{bmatrix}, then find the matrix A−2B+6IA - 2B + 6I, where II is the unit matrix of order 2.
  4. Ex 4.5 Q3
    If A=[12−3−37−80−61]A = \begin{bmatrix} 1 & 2 & -3 \\ -3 & 7 & -8 \\ 0 & -6 & 1 \end{bmatrix}, B=[9−12−42540−3]B = \begin{bmatrix} 9 & -1 & 2 \\ -4 & 2 & 5 \\ 4 & 0 & -3 \end{bmatrix}, then find the matrix CC such that A+B+CA + B + C is a zero matrix.
  5. Ex 4.5 Q4
    If A=[1−23−5−60]A = \begin{bmatrix} 1 & -2 \\ 3 & -5 \\ -6 & 0 \end{bmatrix}, B=[−1−24215]B = \begin{bmatrix} -1 & -2 \\ 4 & 2 \\ 1 & 5 \end{bmatrix} and C=[24−1−4−36]C = \begin{bmatrix} 2 & 4 \\ -1 & -4 \\ -3 & 6 \end{bmatrix}, find the matrix XX such that 3A−4B+5X=C3A - 4B + 5X = C.
  6. Ex 4.5 Q5
    Solve the following equations for XX and YY, if 3X−Y=[1−1−11]3X - Y = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} and X−3Y=[0−10−1]X - 3Y = \begin{bmatrix} 0 & -1 \\ 0 & -1 \end{bmatrix}.
  7. Ex 4.5 Q6
    Find matrices AA and BB, if 2A−B=[6−60−421]2A - B = \begin{bmatrix} 6 & -6 & 0 \\ -4 & 2 & 1 \end{bmatrix} and A−2B=[328−21−7]A - 2B = \begin{bmatrix} 3 & 2 & 8 \\ -2 & 1 & -7 \end{bmatrix}.
  8. Ex 4.5 Q7
    Simplify, cos⁡θ[cos⁡θsin⁡θ−sin⁡θcos⁡θ]+sin⁡θ[sin⁡θ−cos⁡θcos⁡θsin⁡θ]\cos\theta \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} + \sin\theta \begin{bmatrix} \sin\theta & -\cos\theta \\ \cos\theta & \sin\theta \end{bmatrix}
  9. Ex 4.5 Q8
    If A=[i2i−32]A = \begin{bmatrix} i & 2i \\ -3 & 2 \end{bmatrix} and B=[2ii2−3]B = \begin{bmatrix} 2i & i \\ 2 & -3 \end{bmatrix}, where −1=i\sqrt{-1} = i, find A+BA + B and A−BA - B. Show that A+BA + B is a singular. Is A−BA - B a singular ? Justify your answer.
  10. Ex 4.5 Q9
    Find xx and yy, if [2x+y−1134y4]+[−164303]=[3556187]\begin{bmatrix} 2x + y & -1 & 1 \\ 3 & 4y & 4 \end{bmatrix} + \begin{bmatrix} -1 & 6 & 4 \\ 3 & 0 & 3 \end{bmatrix} = \begin{bmatrix} 3 & 5 & 5 \\ 6 & 18 & 7 \end{bmatrix}
  11. Ex 4.5 Q10
    If [2a+b3a−bc+2d2c−d]=[234−1]\begin{bmatrix} 2a + b & 3a - b \\ c + 2d & 2c - d \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix}, find aa, bb, cc and dd.
  12. There are two book shops owned by Suresh and Ganesh. Their sales (in Rupees) for books in three subject – Physics, Chemistry and Mathematics for two months, July and August 2017 are given by two matrices A and B. July sales (in Rupees), Physics Chemistry Mathematics. A=[560067508500665070558905]A = \begin{bmatrix} 5600 & 6750 & 8500 \\ 6650 & 7055 & 8905 \end{bmatrix} First Row Suresh/ Second Row Ganesh August sales (in Rupees), Physics Chemistry Mathematics B=[6650705589057000750010200]B = \begin{bmatrix} 6650 & 7055 & 8905 \\ 7000 & 7500 & 10200 \end{bmatrix} First Row Suresh/ Second Row Ganesh then,
    Ex 4.5 Q11(i)
    Find the increase in sales in Rupees from July to August 2017.
  13. Ex 4.5 Q11(ii)
    If both book shops got 10 % profit in the month of August 2017, find the profit for each book seller in each subject in that month.

4.5 Algebra of Matrices

5 q

Solved Examples (continued)

Worked · 5
  1. 4.5b SolvedEx.1
    If A=[a11a12a13]A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \end{bmatrix} and B=[b11b21b31]B = \begin{bmatrix} b_{11} \\ b_{21} \\ b_{31} \end{bmatrix}, find ABAB.
  2. 4.5b SolvedEx.2
    Let A=[132]1×3A = \begin{bmatrix} 1 & 3 & 2 \end{bmatrix}_{1\times 3} and B=[321]3×1B = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}_{3\times 1}, find ABAB. Does BABA exist? If yes, find it.
  3. 4.5b SolvedEx.3
    A=[−1−2−3210]3×2A = \begin{bmatrix} -1 & -2 \\ -3 & 2 \\ 1 & 0 \end{bmatrix}_{3\times 2}, B=[12−1−2]2×2B = \begin{bmatrix} 1 & 2 \\ -1 & -2 \end{bmatrix}_{2\times 2}. Find ABAB and BABA which ever exist.
  4. 4.5b SolvedEx.4
    Let A=[32−1−254]2×3A = \begin{bmatrix} 3 & 2 & -1 \\ -2 & 5 & 4 \end{bmatrix}_{2\times 3}, B=[3−3−42]2×2B = \begin{bmatrix} 3 & -3 \\ -4 & 2 \end{bmatrix}_{2\times 2}. Find ABAB and BABA which ever exist.
  5. 4.5b SolvedEx.5
    Let A=[4−352]A = \begin{bmatrix} 4 & -3 \\ 5 & 2 \end{bmatrix} and B=[−134−2]B = \begin{bmatrix} -1 & 3 \\ 4 & -2 \end{bmatrix}. Find ABAB and BABA which ever exist.

4.6 Properties of Matrix Multiplication

8 q

Solved Examples

Worked · 8
  1. 4.6 SolvedEx.1
    If A=[1−120−13]A = \begin{bmatrix} 1 & -1 & 2 \\ 0 & -1 & 3 \end{bmatrix}, B=[−213−102]B = \begin{bmatrix} -2 & 1 \\ 3 & -1 \\ 0 & 2 \end{bmatrix}, show that matrix ABAB is non singular.
  2. 4.6 SolvedEx.2
    If A=[133313331]A = \begin{bmatrix} 1 & 3 & 3 \\ 3 & 1 & 3 \\ 3 & 3 & 1 \end{bmatrix} prove that A2−5AA^2 - 5A is a scalar matrix.
  3. 4.6 SolvedEx.3
    If A=[3−24−2]A = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix}, find kk, so that A2−kA+2I=OA^2 - kA + 2I = O, where II is an identity matrix and OO is null matrix of order 2.
  4. 4.6 SolvedEx.4
    Find xx and yy, if [203]{3[63−1254]+2[−4−110−3−4]}=[xy]\begin{bmatrix} 2 & 0 & 3 \end{bmatrix}\left\{ 3\begin{bmatrix} 6 & 3 \\ -1 & 2 \\ 5 & 4 \end{bmatrix} + 2\begin{bmatrix} -4 & -1 \\ 1 & 0 \\ -3 & -4 \end{bmatrix} \right\} = \begin{bmatrix} x & y \end{bmatrix}
  5. 4.6 SolvedEx.5
    Find if [sin⁡θcos⁡θθ][sin⁡θcos⁡θθ]=[17]\begin{bmatrix} \sin\theta \\ \cos\theta \\ \theta \end{bmatrix}\begin{bmatrix} \sin\theta & \cos\theta & \theta \end{bmatrix} = \begin{bmatrix} 17 \end{bmatrix}
  6. 4.6 SolvedEx.6
    If A=[a00b]A = \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}, prove that An=[an00bn]A^n = \begin{bmatrix} a^n & 0 \\ 0 & b^n \end{bmatrix} for all n∈Nn \in N.
  7. 4.6 SolvedEx.7
    A school purchased 8 dozen Mathematics books, 7 dozen Physics books and 10 dozen Chemistry books, the prices are Rs.50, Rs.40 and Rs.60 per book respectively. Find the total amount that the book seller will receive from school authority using matrix multiplication.
  8. 4.6 SolvedEx.8
    Some schools send their students for extra training in Kabaddi, Cricket and Tennis to a sports standidium. There center charge fee is changed pen student for Coaching as well as 4 equipment and maintenances of the court. The information of students from each school is given below-
    KabaddiCricketTennis
    Modern School203515
    Progressive School183612
    Sharada Sadan24128
    Vidya Niketan25206
    The charges per student for each game are given below-
    CoachE & M
    Kabaddi4010
    Cricket5050
    Tennis6040
    E and M is for equipment and maintain. Find the expense of each school on Coaching and on E and M by multiplication of the above matrices.

Exercise 4.6

29 q
  1. Evaluate.
    Ex 4.6 Q1(i)
    [321][2−43]\begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}\begin{bmatrix} 2 & -4 & 3 \end{bmatrix}
  2. Ex 4.6 Q1(ii)
    [2−13][431]\begin{bmatrix} 2 & -1 & 3 \end{bmatrix}\begin{bmatrix} 4 \\ 3 \\ 1 \end{bmatrix}
  3. Ex 4.6 Q2
    If A=[1−342]A = \begin{bmatrix} 1 & -3 \\ 4 & 2 \end{bmatrix}, B=[413−2]B = \begin{bmatrix} 4 & 1 \\ 3 & -2 \end{bmatrix} show that AB≠BAAB \ne BA.
  4. Ex 4.6 Q3
    If A=[−1112301−31]A = \begin{bmatrix} -1 & 1 & 1 \\ 2 & 3 & 0 \\ 1 & -3 & 1 \end{bmatrix}, B=[214302121]B = \begin{bmatrix} 2 & 1 & 4 \\ 3 & 0 & 2 \\ 1 & 2 & 1 \end{bmatrix}. State whether AB=BAAB = BA? Justify your answer.
  5. Show that AB=BAAB = BA where,
    Ex 4.6 Q4(i)
    A=[−23−1−12−1−69−4]A = \begin{bmatrix} -2 & 3 & -1 \\ -1 & 2 & -1 \\ -6 & 9 & -4 \end{bmatrix}, B=[13−122−130−1]B = \begin{bmatrix} 1 & 3 & -1 \\ 2 & 2 & -1 \\ 3 & 0 & -1 \end{bmatrix}
  6. Ex 4.6 Q4(ii)
    A=[cos⁡θsin⁡θsin⁡θcos⁡θ]A = \begin{bmatrix} \cos\theta & \sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}, B=[cos⁡ϕ−sin⁡ϕsin⁡ϕcos⁡ϕ]B = \begin{bmatrix} \cos\phi & -\sin\phi \\ \sin\phi & \cos\phi \end{bmatrix}
  7. Ex 4.6 Q5
    If A=[48−2−4]A = \begin{bmatrix} 4 & 8 \\ -2 & -4 \end{bmatrix}, prove that A2=0A^{2} = 0.
  8. Verify A(BC)=(AB)CA(BC) = (AB)C in each of the following cases.
    Ex 4.6 Q6(i)
    A=[101230045]A = \begin{bmatrix} 1 & 0 & 1 \\ 2 & 3 & 0 \\ 0 & 4 & 5 \end{bmatrix}, B=[2−2−1103]B = \begin{bmatrix} 2 & -2 \\ -1 & 1 \\ 0 & 3 \end{bmatrix} and C=[32−120−2]C = \begin{bmatrix} 3 & 2 & -1 \\ 2 & 0 & -2 \end{bmatrix}
  9. Ex 4.6 Q6(ii)
    A=[243−132]A = \begin{bmatrix} 2 & 4 & 3 \\ -1 & 3 & 2 \end{bmatrix}, B=[2−233−11]B = \begin{bmatrix} 2 & -2 \\ 3 & 3 \\ -1 & 1 \end{bmatrix} and C=[3113]C = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix}
  10. Verify that A(B+C)=AB+BCA(B+C) = AB + BC in each of the following matrices
    Ex 4.6 Q7(i)
    A=[4−223]A = \begin{bmatrix} 4 & -2 \\ 2 & 3 \end{bmatrix}, B=[−113−2]B = \begin{bmatrix} -1 & 1 \\ 3 & -2 \end{bmatrix} and C=[412−1]C = \begin{bmatrix} 4 & 1 \\ 2 & -1 \end{bmatrix}
  11. Ex 4.6 Q7(ii)
    A=[1−13232]A = \begin{bmatrix} 1 & -1 & 3 \\ 2 & 3 & 2 \end{bmatrix}, B=[10−2343]B = \begin{bmatrix} 1 & 0 \\ -2 & 3 \\ 4 & 3 \end{bmatrix} and C=[12−204−3]C = \begin{bmatrix} 1 & 2 \\ -2 & 0 \\ 4 & -3 \end{bmatrix}
  12. Ex 4.6 Q8
    If A=[1−256]A = \begin{bmatrix} 1 & -2 \\ 5 & 6 \end{bmatrix}, B=[3−137]B = \begin{bmatrix} 3 & -1 \\ 3 & 7 \end{bmatrix}, find AB−2IAB - 2I, where II is unit matrix of order 2.
  13. Ex 4.6 Q9
    If A=[432−120]A = \begin{bmatrix} 4 & 3 & 2 \\ -1 & 2 & 0 \end{bmatrix}, B=[12−101−2]B = \begin{bmatrix} 1 & 2 \\ -1 & 0 \\ 1 & -2 \end{bmatrix} show that matrix ABAB is non singular.
  14. Ex 4.6 Q10
    If A=[12054207−3]A = \begin{bmatrix} 1 & 2 & 0 \\ 5 & 4 & 2 \\ 0 & 7 & -3 \end{bmatrix}, find the product (A+I)(A−I)(A+I)(A-I).
  15. Ex 4.6 Q11
    A=[α011]A = \begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix}, B=[1021]B = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} find α\alpha, if A2=BA^{2} = B.
  16. Ex 4.6 Q12
    If A=[122212221]A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}, show that A2−4AA^{2} - 4A is a scalar matrix.
  17. Ex 4.6 Q13
    If A=[10−17]A = \begin{bmatrix} 1 & 0 \\ -1 & 7 \end{bmatrix}, find kk so that A2−8A−kI=OA^{2} - 8A - kI = O, where II is a unit matrix and OO is a null matrix of order 2.
  18. Ex 4.6 Q14
    If A=[84105]A = \begin{bmatrix} 8 & 4 \\ 10 & 5 \end{bmatrix}, B=[5−410−8]B = \begin{bmatrix} 5 & -4 \\ 10 & -8 \end{bmatrix} show that (A+B)2=A2+AB+B2(A + B)^{2} = A^{2} + AB + B^{2}.
  19. Ex 4.6 Q15
    If A=[31−12]A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}, prove that A2−5A+7I=0A^{2} - 5A + 7I = 0, where II is unit matrix of order 2.
  20. Ex 4.6 Q16
    If A=[34−43]A = \begin{bmatrix} 3 & 4 \\ -4 & 3 \end{bmatrix} and B=[21−12]B = \begin{bmatrix} 2 & 1 \\ -1 & 2 \end{bmatrix}, show that (A+B)(A−B)=A2−B2(A+B)(A-B) = A^{2} - B^{2}.
  21. Ex 4.6 Q17
    If A=[12−1−2]A = \begin{bmatrix} 1 & 2 \\ -1 & -2 \end{bmatrix}, B=[2a−1b]B = \begin{bmatrix} 2 & a \\ -1 & b \end{bmatrix} and if (A+B)2=A2+B2(A + B)^{2} = A^{2} + B^{2}, find values of aa and bb.
  22. Ex 4.6 Q18
    Find matrix XX such that AX=BAX = B, where A=[1−2−21]A = \begin{bmatrix} 1 & -2 \\ -2 & 1 \end{bmatrix} and B=[−3−1]B = \begin{bmatrix} -3 \\ -1 \end{bmatrix}.
  23. Ex 4.6 Q19
    Find kk, if A=[3−24−2]A = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix} and if A2=kA−2IA^{2} = kA - 2I.
  24. Ex 4.6 Q20
    Find xx, if [1x1][123456325][1−23]=0\begin{bmatrix} 1 & x & 1 \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5 \end{bmatrix}\begin{bmatrix} 1 \\ -2 \\ 3 \end{bmatrix} = 0.
  25. Ex 4.6 Q21
    Find xx and yy, if {4[2−13102]−[3−34211]}[2−11]=[xy]\left\{ 4\begin{bmatrix} 2 & -1 & 3 \\ 1 & 0 & 2 \end{bmatrix} - \begin{bmatrix} 3 & -3 & 4 \\ 2 & 1 & 1 \end{bmatrix} \right\}\begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix}.
  26. Ex 4.6 Q22
    Find x,y,zx, y, z if {3[200222]−4[11−1231]}[12]=[x−3y−12z]\left\{ 3\begin{bmatrix} 2 & 0 \\ 0 & 2 \\ 2 & 2 \end{bmatrix} - 4\begin{bmatrix} 1 & 1 \\ -1 & 2 \\ 3 & 1 \end{bmatrix} \right\}\begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} x-3 \\ y-1 \\ 2z \end{bmatrix}.
  27. Ex 4.6 Q23
    If A=[cos⁡αsin⁡α−sin⁡αcos⁡α]A = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix}, show that A2=[cos⁡2αsin⁡2α−sin⁡2αcos⁡2α]A^{2} = \begin{bmatrix} \cos 2\alpha & \sin 2\alpha \\ -\sin 2\alpha & \cos 2\alpha \end{bmatrix}.
  28. Ex 4.6 Q24
    If A=[1235]A = \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}, B=[042−1]B = \begin{bmatrix} 0 & 4 \\ 2 & -1 \end{bmatrix}, show that AB≠BAAB \ne BA, but ∣AB∣=∣A∣⋅∣B∣|AB| = |A| \cdot |B|.
  29. Ex 4.6 Q25
    Jay and Ram are two friends in a class. Jay wanted to buy 4 pens and 8 notebooks, Ram wanted to buy 5 pens and 12 notebooks. Both of them went to a shop. The price of a pen and a notebook which they have selected was Rs.6 and Rs.10. Using Matrix multiplication, find the amount required from each one of them.

Exercise 4.7

20 q
  1. Find ATA^{T}, if
    Ex 4.7 Q1(i)
    A=[13−45]A = \begin{bmatrix} 1 & 3 \\ -4 & 5 \end{bmatrix}
  2. Ex 4.7 Q1(ii)
    A=[2−61−405]A = \begin{bmatrix} 2 & -6 & 1 \\ -4 & 0 & 5 \end{bmatrix}
  3. Ex 4.7 Q2
    If [aij]3×3\left[ a_{ij} \right]_{3 \times 3} where aij=2(i−j)a_{ij} = 2(i - j). Find AA and ATA^{T}. State whether AA and ATA^{T} are symmetric or skew symmetric matrices?
  4. Ex 4.7 Q3
    If A=[5−34−3−21]A = \begin{bmatrix} 5 & -3 \\ 4 & -3 \\ -2 & 1 \end{bmatrix}, prove that (2A)T=2AT(2A)^{T} = 2A^{T}.
  5. Ex 4.7 Q4
    If A=[12−52−34−549]A = \begin{bmatrix} 1 & 2 & -5 \\ 2 & -3 & 4 \\ -5 & 4 & 9 \end{bmatrix}, prove that (3A)T=3AT(3A)^{T} = 3A^{T}.
  6. Ex 4.7 Q5
    If A=[01+2ii−2−1−2i0−72−i70]A = \begin{bmatrix} 0 & 1+2i & i-2 \\ -1-2i & 0 & -7 \\ 2-i & 7 & 0 \end{bmatrix} where i=−1i = \sqrt{-1}, prove that AT=−AA^{T} = -A.
  7. If A=[2−35−4−61]A = \begin{bmatrix} 2 & -3 \\ 5 & -4 \\ -6 & 1 \end{bmatrix}, B=[214−1−33]B = \begin{bmatrix} 2 & 1 \\ 4 & -1 \\ -3 & 3 \end{bmatrix} and C=[12−14−23]C = \begin{bmatrix} 1 & 2 \\ -1 & 4 \\ -2 & 3 \end{bmatrix} then show that
    Ex 4.7 Q6(i)
    (A+B)=AT+BT(A + B) = A^{T} + B^{T}
  8. Ex 4.7 Q6(ii)
    (A−C)T=AT−CT(A - C)^{T} = A^{T} - C^{T}
  9. Ex 4.7 Q7
    If A=[54−23]A = \begin{bmatrix} 5 & 4 \\ -2 & 3 \end{bmatrix} and B=[−134−1]B = \begin{bmatrix} -1 & 3 \\ 4 & -1 \end{bmatrix}, then find CTC^{T}, such that 3A−2B+C=I3A - 2B + C = I, where II is the unit matrix of order 2.
  10. If A=[73004−2]A = \begin{bmatrix} 7 & 3 & 0 \\ 0 & 4 & -2 \end{bmatrix}, B=[0−2321−4]B = \begin{bmatrix} 0 & -2 & 3 \\ 2 & 1 & -4 \end{bmatrix} then find
    Ex 4.7 Q8(i)
    AT+4BTA^{T} + 4B^{T}
  11. Ex 4.7 Q8(ii)
    5AT−5BT5A^{T} - 5B^{T}
  12. Ex 4.7 Q9
    If A=[101312]A = \begin{bmatrix} 1 & 0 & 1 \\ 3 & 1 & 2 \end{bmatrix}, B=[21−435−2]B = \begin{bmatrix} 2 & 1 & -4 \\ 3 & 5 & -2 \end{bmatrix} and C=[023−1−10]C = \begin{bmatrix} 0 & 2 & 3 \\ -1 & -1 & 0 \end{bmatrix}, verify that (A+2B+2C)T=AT+2BT+3CT(A + 2B + 2C)^{T} = A^{T} + 2B^{T} + 3C^{T}.
  13. Ex 4.7 Q10
    If A=[−121−32−3]A = \begin{bmatrix} -1 & 2 & 1 \\ -3 & 2 & -3 \end{bmatrix} and B=[21−32−13]B = \begin{bmatrix} 2 & 1 \\ -3 & 2 \\ -1 & 3 \end{bmatrix}, prove that (A+BT)T=AT+B(A + B^{T})^{T} = A^{T} + B.
  14. Prove that A+ATA + A^{T} is a symmetric and A−ATA - A^{T} is a skew symmetric matrix, where
    Ex 4.7 Q11(i)
    A=[124321−2−32]A = \begin{bmatrix} 1 & 2 & 4 \\ 3 & 2 & 1 \\ -2 & -3 & 2 \end{bmatrix}
  15. Ex 4.7 Q11(ii)
    A=[52−43−724−5−3]A = \begin{bmatrix} 5 & 2 & -4 \\ 3 & -7 & 2 \\ 4 & -5 & -3 \end{bmatrix}
  16. Express the following matrices as the sum of a symmetric and a skew symmetric matrix.
    Ex 4.7 Q12(i)
    [4−23−5]\begin{bmatrix} 4 & -2 \\ 3 & -5 \end{bmatrix}
  17. Ex 4.7 Q12(ii)
    [33−1−2−21−4−52]\begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix}
  18. If A=[2−13−241]A = \begin{bmatrix} 2 & -1 \\ 3 & -2 \\ 4 & 1 \end{bmatrix} and B=[03−42−11]B = \begin{bmatrix} 0 & 3 & -4 \\ 2 & -1 & 1 \end{bmatrix}, verify that
    Ex 4.7 Q13(i)
    (AB)T=BTAT(AB)^{T} = B^{T}A^{T}
  19. Ex 4.7 Q13(ii)
    (BA)T=ATBT(BA)^{T} = A^{T}B^{T}
  20. Ex 4.7 Q14
    If A=[cos⁡αsin⁡α−sin⁡αcos⁡α]A = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix}, show that ATA=IA^{T}A = I, where II is the unit matrix of order 2.

Miscellaneous Exercise 4 (B)

42 q

(I) Select the correct option

Practice · 10
  1. Misc 4B I Q1
    Given A=[1322]A = \begin{bmatrix} 1 & 3 \\ 2 & 2 \end{bmatrix}, I=[1001]I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} if A−λIA - \lambda I is a singular matrix then ..........

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  2. Misc 4B I Q2
    Consider the matrices A=[46−13021−25]A = \begin{bmatrix} 4 & 6 & -1 \\ 3 & 0 & 2 \\ 1 & -2 & 5 \end{bmatrix}, B=[2401−12]B = \begin{bmatrix} 2 & 4 \\ 0 & 1 \\ -1 & 2 \end{bmatrix}, C=[312]C = \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix} out of the given matrix product ........... i) (AB)TC(AB)^{T}C ii) CTC(AB)TC^{T}C(AB)^{T} iii) CTABC^{T}AB iv) ATABBTCA^{T}ABB^{T}C

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  3. Misc 4B I Q3
    If AA and BB are square matrices of equal order, then which one is correct among the following?

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  4. Misc 4B I Q4
    If A=[12221−2a2b]A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix} is a matrix satisfying the equation AAT=9IAA^{T} = 9I, where II is the identity matrix of order 3, then the ordered pair (a,b)(a, b) is equal to .......

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  5. Misc 4B I Q5
    If A=[α22α]A = \begin{bmatrix} \alpha & 2 \\ 2 & \alpha \end{bmatrix} and ∣A3∣=125|A^{3}| = 125, then α=\alpha = .......

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  6. Misc 4B I Q6
    If [57x126]−[12−352y]=[454−404]\begin{bmatrix} 5 & 7 \\ x & 1 \\ 2 & 6 \end{bmatrix} - \begin{bmatrix} 1 & 2 \\ -3 & 5 \\ 2 & y \end{bmatrix} = \begin{bmatrix} 4 & 5 \\ 4 & -4 \\ 0 & 4 \end{bmatrix} then ......

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  7. Misc 4B I Q7
    If A+B=[7489]A + B = \begin{bmatrix} 7 & 4 \\ 8 & 9 \end{bmatrix} and A−B=[1203]A - B = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix} then the value of AA is .......

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  8. Misc 4B I Q8
    If [x3x−yzx+z3y−w]=[3247]\begin{bmatrix} x & 3x - y \\ zx + z & 3y - w \end{bmatrix} = \begin{bmatrix} 3 & 2 \\ 4 & 7 \end{bmatrix} then ..........

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  9. Misc 4B I Q9
    For suitable matrices A,BA, B, the false statement is .......

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  10. Misc 4B I Q10
    If A=[−2103]A = \begin{bmatrix} -2 & 1 \\ 0 & 3 \end{bmatrix} and f(x)=2x2−3xf(x) = 2x^{2} - 3x, then f(A)=f(A) = .........

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(II) Answer the following questions

Practice · 32
  1. If A=diag [2  −3  −5]A = \text{diag}\,[2\ \ -3\ \ -5], B=diag [4  −6  −3]B = \text{diag}\,[4\ \ -6\ \ -3] and C=diag [−3  4  1]C = \text{diag}\,[-3\ \ 4\ \ 1] then find
    Misc 4B II Q1(i)
    B+C−AB + C - A
  2. Misc 4B II Q1(ii)
    2A+B−5C2A + B - 5C
  3. If f(α)=A=[cos⁡α−sin⁡α0sin⁡αcos⁡α0001]f(\alpha) = A = \begin{bmatrix} \cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}, find
    Misc 4B II Q2(i)
    f(−α)f(-\alpha)
  4. Misc 4B II Q2(ii)
    f(−α)+f(α)f(-\alpha) + f(\alpha)
  5. Find matrices AA and BB, where
    Misc 4B II Q3(i)
    2A−B=[1−101]2A - B = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix} and A+3B=[1−101]A + 3B = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}
  6. Misc 4B II Q3(ii)
    3A−B=[−121105]3A - B = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 0 & 5 \end{bmatrix} and A+5B=[001−100]A + 5B = \begin{bmatrix} 0 & 0 & 1 \\ -1 & 0 & 0 \end{bmatrix}
  7. If A=[2−33−2−14]A = \begin{bmatrix} 2 & -3 \\ 3 & -2 \\ -1 & 4 \end{bmatrix}, B=[−3412−1−3]B = \begin{bmatrix} -3 & 4 & 1 \\ 2 & -1 & -3 \end{bmatrix}, verify
    Misc 4B II Q4(i)
    (A+BT)T=AT+2B(A + B^{T})^{T} = A^{T} + 2B
  8. Misc 4B II Q4(ii)
    (3A−5BT)T=3AT−5B(3A - 5B^{T})^{T} = 3A^{T} - 5B
  9. Misc 4B II Q5
    If A=[cos⁡α−sin⁡αsin⁡αcos⁡α]A = \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix} and A+AT=IA + A^{T} = I, where II is unit matrix 2×22 \times 2, then find the value of α\alpha.
  10. Misc 4B II Q6
    If A=[1232−10]A = \begin{bmatrix} 1 & 2 \\ 3 & 2 \\ -1 & 0 \end{bmatrix} and B=[1324−1−3]B = \begin{bmatrix} 1 & 3 & 2 \\ 4 & -1 & -3 \end{bmatrix}, show that ABAB is singular.
  11. Misc 4B II Q7
    If A=[123246123]A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 2 & 3 \end{bmatrix}, B=[1−11−32−1−210]B = \begin{bmatrix} 1 & -1 & 1 \\ -3 & 2 & -1 \\ -2 & 1 & 0 \end{bmatrix}, show that ABAB and BABA are both singular matrices.
  12. Misc 4B II Q8
    If A=[1−10234012]A = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix}, B=[22−4−42−42−15]B = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{bmatrix}, show that BA=6IBA = 6I.
  13. Misc 4B II Q9
    If A=[2103]A = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}, B=[123−2]B = \begin{bmatrix} 1 & 2 \\ 3 & -2 \end{bmatrix}, verify that ∣AB∣=∣A∣∣B∣|AB| = |A||B|.
  14. Misc 4B II Q10
    If Aα=[cos⁡αsin⁡α−sin⁡αcos⁡α]A_{\alpha} = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix}, show that Aα⋅Aβ=Aα+βA_{\alpha} \cdot A_{\beta} = A_{\alpha + \beta}.
  15. Misc 4B II Q11
    If A=[1ωω21]A = \begin{bmatrix} 1 & \omega \\ \omega^{2} & 1 \end{bmatrix}, B=[ω211ω]B = \begin{bmatrix} \omega^{2} & 1 \\ 1 & \omega \end{bmatrix}, where ω\omega is a complex cube root of unity, then show that AB+BA+A−2BAB + BA + A - 2B is a null matrix.
  16. Misc 4B II Q12
    If A=[2−2−4−1341−2−3]A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix} show that A2=AA^{2} = A.
  17. Misc 4B II Q13
    If A=[4−1−430−43−1−3]A = \begin{bmatrix} 4 & -1 & -4 \\ 3 & 0 & -4 \\ 3 & -1 & -3 \end{bmatrix}, show that A2=IA^{2} = I.
  18. Misc 4B II Q14
    If A=[3−5−42]A = \begin{bmatrix} 3 & -5 \\ -4 & 2 \end{bmatrix}, show that A2−5A−14I=0A^{2} - 5A - 14I = 0.
  19. Misc 4B II Q15
    If A=[2−1−12]A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}, show that A2−4A+3I=0A^{2} - 4A + 3I = 0.
  20. Misc 4B II Q16
    If A=[−322−4]A = \begin{bmatrix} -3 & 2 \\ 2 & -4 \end{bmatrix}, B=[1xy0]B = \begin{bmatrix} 1 & x \\ y & 0 \end{bmatrix}, and (A+B)(A−B)=A2−B2(A + B)(A - B) = A^{2} - B^{2}, find xx and yy.
  21. Misc 4B II Q17
    If A=[0110]A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} and B=[0−110]B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} show that (A+B)(A−B)≠A2−B2(A + B)(A - B) \ne A^{2} - B^{2}.
  22. Misc 4B II Q18
    If A=[2−13−2]A = \begin{bmatrix} 2 & -1 \\ 3 & -2 \end{bmatrix}, find A3A^{3}.
  23. Find x,yx, y if,
    Misc 4B II Q19(i)
    [0−14]{2[45362−1]+3[43140−1]}=[xy]\begin{bmatrix} 0 & -1 & 4 \end{bmatrix}\left\{ 2\begin{bmatrix} 4 & 5 \\ 3 & 6 \\ 2 & -1 \end{bmatrix} + 3\begin{bmatrix} 4 & 3 \\ 1 & 4 \\ 0 & -1 \end{bmatrix} \right\} = \begin{bmatrix} x & y \end{bmatrix}
  24. Misc 4B II Q19(ii)
    {−1[121203]+3[2−371−13]}[50−1]=[xy]\left\{ -1\begin{bmatrix} 1 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} + 3\begin{bmatrix} 2 & -3 & 7 \\ 1 & -1 & 3 \end{bmatrix} \right\}\begin{bmatrix} 5 \\ 0 \\ -1 \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix}
  25. Find x,y,zx, y, z if
    Misc 4B II Q20(i)
    {5[011011]−3[213−213]}[21]=[x−1y+12z]\left\{ 5\begin{bmatrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{bmatrix} - 3\begin{bmatrix} 2 & 1 \\ 3 & -2 \\ 1 & 3 \end{bmatrix} \right\}\begin{bmatrix} 2 \\ 1 \end{bmatrix} = \begin{bmatrix} x-1 \\ y+1 \\ 2z \end{bmatrix}
  26. Misc 4B II Q20(ii)
    {[132201312]+2[302145210]}[123]=[xyz]\left\{ \begin{bmatrix} 1 & 3 & 2 \\ 2 & 0 & 1 \\ 3 & 1 & 2 \end{bmatrix} + 2\begin{bmatrix} 3 & 0 & 2 \\ 1 & 4 & 5 \\ 2 & 1 & 0 \end{bmatrix} \right\}\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}
  27. Misc 4B II Q21
    If A=[21−3026]A = \begin{bmatrix} 2 & 1 & -3 \\ 0 & 2 & 6 \end{bmatrix}, B=[10−23−14]B = \begin{bmatrix} 1 & 0 & -2 \\ 3 & -1 & 4 \end{bmatrix}, find ABTAB^{T} and ATBA^{T}B.
  28. Misc 4B II Q22
    If A=[2−43−201]A = \begin{bmatrix} 2 & -4 \\ 3 & -2 \\ 0 & 1 \end{bmatrix}, B=[1−12−210]B = \begin{bmatrix} 1 & -1 & 2 \\ -2 & 1 & 0 \end{bmatrix}, show that (AB)T=BTAT(AB)^{T} = B^{T}A^{T}.
  29. Misc 4B II Q23
    If A=[3−41−1]A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}, prove that An=[1+2n−4nn1−2n]A^{n} = \begin{bmatrix} 1 + 2n & -4n \\ n & 1 - 2n \end{bmatrix}, for all n∈Nn \in N.
  30. Misc 4B II Q24
    If A=[cos⁡θsin⁡θ−sin⁡θcos⁡θ]A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}, prove that An=[cos⁡nθsin⁡nθ−sin⁡nθcos⁡nθ]A^{n} = \begin{bmatrix} \cos n\theta & \sin n\theta \\ -\sin n\theta & \cos n\theta \end{bmatrix}, for all n∈Nn \in N.
  31. Two farmers Shantaram and Kantaram cultivate three crops rice, wheat and groundnut. The sale (In Rupees) of these crops by both the farmers for the month of April and may 2008 is given below, April sale (In Rs.)
    RiceWheatGroundnut
    Shantaram150001300012000
    Kantaram18000150008000
    May sale (In Rs.)
    RiceWheatGroundnut
    Shantaram180001500012000
    Kantaram210001650016000
    Find
    Misc 4B II Q25(i)
    The total sale in rupees for two months of each farmer for each crop.
  32. Misc 4B II Q25(ii)
    the increase in sale from April to May for every crop of each farmer.