Mathematics · Textbook solutions

Matrices

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 115 questions

3.2 Matrices and 3.3 Types of Matrices

22 q

Solved Examples

Worked · 5
  1. 3.1 Eg.1
    Consider the following information regarding the number of men and women workers in three factories I, II and III
    FactoryMen workersWomen workers
    I3025
    II2531
    III2726
    Represent the above information in the form of a 3×23 \times 2 matrix. What does the entry in the third row and second column represent?
  2. 3.1 Eg.2
    If a matrix has 8 elements, what are the possible orders it can have?
  3. 3.1 Eg.3
    Construct a 3×23 \times 2 matrix whose elements are given by aij=12∣i−3j∣a_{ij} = \frac{1}{2}|i - 3j|.
  4. 3.1 Eg.4
    If [x+3z+42y−7−6a−10b−3−210]=[063y−2−6−32c+22b+4−210]\begin{bmatrix} x+3 & z+4 & 2y-7 \\ -6 & a-1 & 0 \\ b-3 & -21 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 6 & 3y-2 \\ -6 & -3 & 2c+2 \\ 2b+4 & -21 & 0 \end{bmatrix} Find the values of aa, bb, cc, xx, yy and zz.
  5. 3.1 Eg.5
    Find the values of aa, bb, cc, and dd from the following equation: [2a+ba−2b5c−d4c+3d]=[4−31124]\begin{bmatrix} 2a+b & a-2b \\ 5c-d & 4c+3d \end{bmatrix} = \begin{bmatrix} 4 & -3 \\ 11 & 24 \end{bmatrix}

Exercise 3.1

Practice · 17
  1. In the matrix A=[2519−735−2521231−517]A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}, write:
    Ex 3.1 Q1 (i)
    The order of the matrix,
  2. Ex 3.1 Q1 (ii)
    The number of elements,
  3. Ex 3.1 Q1 (iii)
    Write the elements a13a_{13}, a21a_{21}, a33a_{33}, a24a_{24}, a23a_{23}.
  4. Ex 3.1 Q2
    If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?
  5. Ex 3.1 Q3
    If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
  6. Construct a 2×22 \times 2 matrix, A=[aij]A = [a_{ij}], whose elements are given by:
    Ex 3.1 Q4 (i)
    aij=(i+j)22a_{ij} = \frac{(i+j)^2}{2}
  7. Ex 3.1 Q4 (ii)
    aij=ija_{ij} = \frac{i}{j}
  8. Ex 3.1 Q4 (iii)
    aij=(i+2j)22a_{ij} = \frac{(i+2j)^2}{2}
  9. Construct a 3×43 \times 4 matrix, whose elements are given by:
    Ex 3.1 Q5 (i)
    aij=12∣−3i+j∣a_{ij} = \frac{1}{2}|-3i + j|
  10. Ex 3.1 Q5 (ii)
    aij=2i−ja_{ij} = 2i - j
  11. Find the values of xx, yy and zz from the following equations:
    Ex 3.1 Q6 (i)
    [43x5]=[yz15]\begin{bmatrix} 4 & 3 \\ x & 5 \end{bmatrix} = \begin{bmatrix} y & z \\ 1 & 5 \end{bmatrix}
  12. Ex 3.1 Q6 (ii)
    [x+y25+zxy]=[6258]\begin{bmatrix} x+y & 2 \\ 5+z & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}
  13. Ex 3.1 Q6 (iii)
    [x+y+zx+zy+z]=[957]\begin{bmatrix} x+y+z \\ x+z \\ y+z \end{bmatrix} = \begin{bmatrix} 9 \\ 5 \\ 7 \end{bmatrix}
  14. Ex 3.1 Q7
    Find the value of aa, bb, cc and dd from the equation: [a−b2a+c2a−b3c+d]=[−15013]\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}
  15. Ex 3.1 Q8
    A=[aij]m×nA = [a_{ij}]_{m \times n} is a square matrix, if
    1. A.
      m<nm < n
    2. B.
      m>nm > n
    3. C.
      m=nm = n
    4. D.
      None of these
  16. Ex 3.1 Q9
    Which of the given values of xx and yy make the following pair of matrices equal [3x+75y+12−3x]\begin{bmatrix} 3x+7 & 5 \\ y+1 & 2-3x \end{bmatrix}, [0y−284]\begin{bmatrix} 0 & y-2 \\ 8 & 4 \end{bmatrix}
    1. A.
      x=−13, y=7x = \dfrac{-1}{3},\ y = 7
    2. B.
      Not possible to find
    3. C.
      y=7, x=−23y = 7,\ x = \dfrac{-2}{3}
    4. D.
      x=−13, y=−23x = \dfrac{-1}{3},\ y = \dfrac{-2}{3}
  17. Ex 3.1 Q10
    The number of all possible matrices of order 3×33 \times 3 with each entry 0 or 1 is:
    1. A.
      27
    2. B.
      18
    3. C.
      81
    4. D.
      512

3.4 Operations on Matrices

51 q

Solved Examples

Worked · 14
  1. 3.2 Eg.6
    Given A=[31−1230]A = \begin{bmatrix} \sqrt{3} & 1 & -1 \\ 2 & 3 & 0 \end{bmatrix} and B=[251−2312]B = \begin{bmatrix} 2 & \sqrt{5} & 1 \\ -2 & 3 & \frac{1}{2} \end{bmatrix}, find A+BA + B.
  2. 3.2 Eg.7
    If A=[123231]A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{bmatrix} and B=[3−13−102]B = \begin{bmatrix} 3 & -1 & 3 \\ -1 & 0 & 2 \end{bmatrix}, then find 2A−B2A - B.
  3. 3.2 Eg.8
    If A=[804−236]A = \begin{bmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{bmatrix} and B=[2−242−51]B = \begin{bmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{bmatrix}, then find the matrix XX, such that 2A+3X=5B2A + 3X = 5B.
  4. 3.2 Eg.9
    Find XX and YY, if X+Y=[5209]X + Y = \begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix} and X−Y=[360−1]X - Y = \begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix}.
  5. 3.2 Eg.10
    Find the values of xx and yy from the following equation:
    2[x57y−3]+[3−412]=[761514]2\begin{bmatrix} x & 5 \\ 7 & y - 3 \end{bmatrix} + \begin{bmatrix} 3 & -4 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 7 & 6 \\ 15 & 14 \end{bmatrix}
  6. 3.2 Eg.11
    Two farmers Ramkishan and Gurcharan Singh cultivates only three varieties of rice namely Basmati, Permal and Naura. The sale (in Rupees) of these varieties of rice by both the farmers in the month of September and October are given by the following matrices AA and BB. September Sales (in Rupees): A=[10,00020,00030,00050,00030,00010,000]A = \begin{bmatrix} 10{,}000 & 20{,}000 & 30{,}000 \\ 50{,}000 & 30{,}000 & 10{,}000 \end{bmatrix} (row 1 Ramkishan, row 2 Gurcharan Singh; columns Basmati, Permal, Naura). October Sales (in Rupees): B=[500010,000600020,00010,00010,000]B = \begin{bmatrix} 5000 & 10{,}000 & 6000 \\ 20{,}000 & 10{,}000 & 10{,}000 \end{bmatrix} (row 1 Ramkishan, row 2 Gurcharan Singh; columns Basmati, Permal, Naura). (i) Find the combined sales in September and October for each farmer in each variety. (ii) Find the decrease in sales from September to October. (iii) If both farmers receive 2% profit on gross sales, compute the profit for each farmer and for each variety sold in October.
  7. 3.2 Eg.12
    Find AB, if A=[6923]A = \begin{bmatrix} 6 & 9 \\ 2 & 3 \end{bmatrix} and B=[260798]B = \begin{bmatrix} 2 & 6 & 0 \\ 7 & 9 & 8 \end{bmatrix}.
  8. 3.2 Eg.13
    If A=[1−23−425]A = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{bmatrix} and B=[234521]B = \begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{bmatrix}, then find AB, BA. Show that AB≠BAAB \ne BA.
  9. 3.2 Eg.14
    If A=[100−1]A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} and B=[0110]B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, then AB=[01−10]AB = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}.
  10. 3.2 Eg.15
    Find AB, if A=[0−102]A = \begin{bmatrix} 0 & -1 \\ 0 & 2 \end{bmatrix} and B=[3500]B = \begin{bmatrix} 3 & 5 \\ 0 & 0 \end{bmatrix}.
  11. 3.2 Eg.16
    If A=[11−12033−12]A = \begin{bmatrix} 1 & 1 & -1 \\ 2 & 0 & 3 \\ 3 & -1 & 2 \end{bmatrix}, B=[1302−14]B = \begin{bmatrix} 1 & 3 \\ 0 & 2 \\ -1 & 4 \end{bmatrix} and C=[123−420−21]C = \begin{bmatrix} 1 & 2 & 3 & -4 \\ 2 & 0 & -2 & 1 \end{bmatrix}, find A(BC), (AB)C and show that (AB)C = A(BC).
  12. 3.2 Eg.17
    If A=[067−6087−80]A = \begin{bmatrix} 0 & 6 & 7 \\ -6 & 0 & 8 \\ 7 & -8 & 0 \end{bmatrix}, B=[011102120]B = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0 \end{bmatrix}, C=[2−23]C = \begin{bmatrix} 2 \\ -2 \\ 3 \end{bmatrix}. Calculate AC, BC and (A + B)C. Also, verify that (A + B)C = AC + BC.
  13. 3.2 Eg.18
    If A=[1233−21421]A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & -2 & 1 \\ 4 & 2 & 1 \end{bmatrix}, then show that A3−23A−40I=OA^3 - 23A - 40I = O.
  14. 3.2 Eg.19
    In a legislative assembly election, a political group hired a public relations firm to promote its candidate in three ways: telephone, house calls, and letters. The cost per contact (in paise) is given in matrix A as A=[4010050]TelephoneHousecallLetterA = \begin{bmatrix} 40 \\ 100 \\ 50 \end{bmatrix}\begin{matrix} \text{Telephone} \\ \text{Housecall} \\ \text{Letter} \end{matrix} The number of contacts of each type made in two cities X and Y is given by B=[100050050003000100010,000]→X→YB = \begin{bmatrix} 1000 & 500 & 5000 \\ 3000 & 1000 & 10{,}000 \end{bmatrix}\begin{matrix} \rightarrow \text{X} \\ \rightarrow \text{Y} \end{matrix}, where the columns are Telephone, Housecall and Letter respectively. Find the total amount spent by the group in the two cities X and Y.

Exercise 3.2

Practice · 37
  1. Let A=[2432]A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}, B=[13−25]B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}, C=[−2534]C = \begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix}. Find each of the following:
    Ex 3.2 Q1 (i)
    A+BA + B
  2. Ex 3.2 Q1 (ii)
    A−BA - B
  3. Ex 3.2 Q1 (iii)
    3A−C3A - C
  4. Ex 3.2 Q1 (iv)
    ABAB
  5. Ex 3.2 Q1 (v)
    BABA
  6. Compute the following:
    Ex 3.2 Q2 (i)
    [ab−ba]+[abba]\begin{bmatrix} a & b \\ -b & a \end{bmatrix} + \begin{bmatrix} a & b \\ b & a \end{bmatrix}
  7. Ex 3.2 Q2 (ii)
    [a2+b2b2+c2a2+c2a2+b2]+[2ab2bc−2ac−2ab]\begin{bmatrix} a^2+b^2 & b^2+c^2 \\ a^2+c^2 & a^2+b^2 \end{bmatrix} + \begin{bmatrix} 2ab & 2bc \\ -2ac & -2ab \end{bmatrix}
  8. Ex 3.2 Q2 (iii)
    [−14−68516285]+[1276805324]\begin{bmatrix} -1 & 4 & -6 \\ 8 & 5 & 16 \\ 2 & 8 & 5 \end{bmatrix} + \begin{bmatrix} 12 & 7 & 6 \\ 8 & 0 & 5 \\ 3 & 2 & 4 \end{bmatrix}
  9. Ex 3.2 Q2 (iv)
    [cos⁡2xsin⁡2xsin⁡2xcos⁡2x]+[sin⁡2xcos⁡2xcos⁡2xsin⁡2x]\begin{bmatrix} \cos^2 x & \sin^2 x \\ \sin^2 x & \cos^2 x \end{bmatrix} + \begin{bmatrix} \sin^2 x & \cos^2 x \\ \cos^2 x & \sin^2 x \end{bmatrix}
  10. Compute the indicated products.
    Ex 3.2 Q3 (i)
    [ab−ba][a−bba]\begin{bmatrix} a & b \\ -b & a \end{bmatrix}\begin{bmatrix} a & -b \\ b & a \end{bmatrix}
  11. Ex 3.2 Q3 (ii)
    [123][234]\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}\begin{bmatrix} 2 & 3 & 4 \end{bmatrix}
  12. Ex 3.2 Q3 (iii)
    [1−223][123231]\begin{bmatrix} 1 & -2 \\ 2 & 3 \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{bmatrix}
  13. Ex 3.2 Q3 (iv)
    [234345456][1−35024305]\begin{bmatrix} 2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{bmatrix}\begin{bmatrix} 1 & -3 & 5 \\ 0 & 2 & 4 \\ 3 & 0 & 5 \end{bmatrix}
  14. Ex 3.2 Q3 (v)
    [2132−11][101−121]\begin{bmatrix} 2 & 1 \\ 3 & 2 \\ -1 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 & 1 \\ -1 & 2 & 1 \end{bmatrix}
  15. Ex 3.2 Q3 (vi)
    [3−13−102][2−31031]\begin{bmatrix} 3 & -1 & 3 \\ -1 & 0 & 2 \end{bmatrix}\begin{bmatrix} 2 & -3 \\ 1 & 0 \\ 3 & 1 \end{bmatrix}
  16. Ex 3.2 Q4
    If A=[12−35021−11]A = \begin{bmatrix} 1 & 2 & -3 \\ 5 & 0 & 2 \\ 1 & -1 & 1 \end{bmatrix}, B=[3−12425203]B = \begin{bmatrix} 3 & -1 & 2 \\ 4 & 2 & 5 \\ 2 & 0 & 3 \end{bmatrix} and C=[4120321−23]C = \begin{bmatrix} 4 & 1 & 2 \\ 0 & 3 & 2 \\ 1 & -2 & 3 \end{bmatrix}, then compute (A+B)(A+B) and (B−C)(B - C). Also, verify that A+(B−C)=(A+B)−CA + (B - C) = (A + B) - C.
  17. Ex 3.2 Q5
    If A=[2315313234373223]A = \begin{bmatrix} \frac{2}{3} & 1 & \frac{5}{3} \\ \frac{1}{3} & \frac{2}{3} & \frac{4}{3} \\ \frac{7}{3} & 2 & \frac{2}{3} \end{bmatrix} and B=[25351152545756525]B = \begin{bmatrix} \frac{2}{5} & \frac{3}{5} & 1 \\ \frac{1}{5} & \frac{2}{5} & \frac{4}{5} \\ \frac{7}{5} & \frac{6}{5} & \frac{2}{5} \end{bmatrix}, then compute 3A−5B3A - 5B.
  18. Ex 3.2 Q6
    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}
  19. Find X and Y, if
    Ex 3.2 Q7 (i)
    X+Y=[7025]X + Y = \begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix} and X−Y=[3003]X - Y = \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}
  20. Ex 3.2 Q7 (ii)
    2X+3Y=[2340]2X + 3Y = \begin{bmatrix} 2 & 3 \\ 4 & 0 \end{bmatrix} and 3X+2Y=[2−2−15]3X + 2Y = \begin{bmatrix} 2 & -2 \\ -1 & 5 \end{bmatrix}
  21. Ex 3.2 Q8
    Find X, if Y=[3214]Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix} and 2X+Y=[10−32]2X + Y = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix}.
  22. Ex 3.2 Q9
    Find xx and yy, if 2[130x]+[y012]=[5618]2\begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}.
  23. Ex 3.2 Q10
    Solve the equation for x,y,zx, y, z and tt, if 2[xzyt]+3[1−102]=3[3546]2\begin{bmatrix} x & z \\ y & t \end{bmatrix} + 3\begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix} = 3\begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix}.
  24. Ex 3.2 Q11
    If x[23]+y[−11]=[105]x\begin{bmatrix} 2 \\ 3 \end{bmatrix} + y\begin{bmatrix} -1 \\ 1 \end{bmatrix} = \begin{bmatrix} 10 \\ 5 \end{bmatrix}, find the values of xx and yy.
  25. Ex 3.2 Q12
    Given 3[xyzw]=[x6−12w]+[4x+yz+w3]3\begin{bmatrix} x & y \\ z & w \end{bmatrix} = \begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix} + \begin{bmatrix} 4 & x+y \\ z+w & 3 \end{bmatrix}, find the values of x,y,zx, y, z and ww.
  26. Ex 3.2 Q13
    If F(x)=[cos⁡x−sin⁡x0sin⁡xcos⁡x0001]F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}, show that F(x) F(y)=F(x+y)F(x)\,F(y) = F(x + y).
  27. Show that
    Ex 3.2 Q14 (i)
    [5−167][2134]≠[2134][5−167]\begin{bmatrix} 5 & -1 \\ 6 & 7 \end{bmatrix}\begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix} \neq \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} 5 & -1 \\ 6 & 7 \end{bmatrix}
  28. Ex 3.2 Q14 (ii)
    [123010110][−1100−11234]≠[−1100−11234][123010110]\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{bmatrix}\begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4 \end{bmatrix} \neq \begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4 \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{bmatrix}
  29. Ex 3.2 Q15
    Find A2−5A+6IA^2 - 5A + 6I, if A=[2012131−10]A = \begin{bmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{bmatrix}.
  30. Ex 3.2 Q16
    If A=[102021203]A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix}, prove that A3−6A2+7A+2I=0A^3 - 6A^2 + 7A + 2I = 0.
  31. Ex 3.2 Q17
    If A=[3−24−2]A = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix} and I=[1001]I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, find kk so that A2=kA−2IA^2 = kA - 2I.
  32. Ex 3.2 Q18
    If A=[0−tan⁡α2tan⁡α20]A = \begin{bmatrix} 0 & -\tan\frac{\alpha}{2} \\ \tan\frac{\alpha}{2} & 0 \end{bmatrix} and II is the identity matrix of order 2, show that I+A=(I−A)[cos⁡α−sin⁡αsin⁡αcos⁡α]I + A = (I - A)\begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}.
  33. A trust fund has ₹30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide ₹30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:
    Ex 3.2 Q19 (a)
    ₹1800
  34. Ex 3.2 Q19 (b)
    ₹2000
  35. Ex 3.2 Q20
    The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are ₹80, ₹60 and ₹40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.
  36. Ex 3.2 Q21
    Assume X, Y, Z, W and P are matrices of order 2×n2 \times n, 3×k3 \times k, 2×p2 \times p, n×3n \times 3 and p×kp \times k, respectively. The restriction on nn, kk and pp so that PY+WYPY + WY will be defined are:
    1. A.
      k=3k = 3, p=np = n
    2. B.
      kk is arbitrary, p=2p = 2
    3. C.
      pp is arbitrary, k=3k = 3
    4. D.
      k=2k = 2, p=3p = 3
  37. Ex 3.2 Q22
    Assume X, Y, Z, W and P are matrices of order 2×n2 \times n, 3×k3 \times k, 2×p2 \times p, n×3n \times 3 and p×kp \times k, respectively. If n=pn = p, then the order of the matrix 7X−5Z7X - 5Z is:
    1. A.
      p×2p \times 2
    2. B.
      2×n2 \times n
    3. C.
      n×3n \times 3
    4. D.
      p×np \times n

3.5 Transpose and 3.6 Symmetric and Skew Symmetric Matrices

26 q

Solved Examples

Worked · 3
  1. 3.3 Eg.20
    If A=[332420]A = \begin{bmatrix} 3 & \sqrt{3} & 2 \\ 4 & 2 & 0 \end{bmatrix} and B=[2−12124]B = \begin{bmatrix} 2 & -1 & 2 \\ 1 & 2 & 4 \end{bmatrix}, verify that (i) (A′)′=A(A')' = A, (ii) (A+B)′=A′+B′(A + B)' = A' + B', (iii) (kB)′=kB′(kB)' = kB', where kk is any constant.
  2. 3.3 Eg.21
    If A=[−245]A = \begin{bmatrix} -2 \\ 4 \\ 5 \end{bmatrix}, B=[13−6]B = \begin{bmatrix} 1 & 3 & -6 \end{bmatrix}, verify that (AB)′=B′A′(AB)' = B'A'.
  3. 3.3 Eg.22
    Express the matrix B=[2−2−4−1341−2−3]B = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix} as the sum of a symmetric and a skew symmetric matrix.

Exercise 3.3

Practice · 23
  1. Find the transpose of each of the following matrices:
    Ex 3.3 Q1 (i)
    [512−1]\begin{bmatrix} 5 \\ \frac{1}{2} \\ -1 \end{bmatrix}
  2. Ex 3.3 Q1 (ii)
    [1−123]\begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}
  3. Ex 3.3 Q1 (iii)
    [−15635623−1]\begin{bmatrix} -1 & 5 & 6 \\ \sqrt{3} & 5 & 6 \\ 2 & 3 & -1 \end{bmatrix}
  4. If A=[−123579−211]A = \begin{bmatrix} -1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1 \end{bmatrix} and B=[−41−5120131]B = \begin{bmatrix} -4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{bmatrix}, then verify that
    Ex 3.3 Q2 (i)
    (A+B)′=A′+B′(A + B)' = A' + B'
  5. Ex 3.3 Q2 (ii)
    (A−B)′=A′−B′(A - B)' = A' - B'
  6. If A′=[34−1201]A' = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix} and B=[−121123]B = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}, then verify that
    Ex 3.3 Q3 (i)
    (A+B)′=A′+B′(A + B)' = A' + B'
  7. Ex 3.3 Q3 (ii)
    (A−B)′=A′−B′(A - B)' = A' - B'
  8. Ex 3.3 Q4
    If A′=[−2312]A' = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix} and B=[−1012]B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}, then find (A+2B)′(A + 2B)'.
  9. For the matrices A and B, verify that (AB)′=B′A′(AB)' = B'A', where
    Ex 3.3 Q5 (i)
    A=[1−43]A = \begin{bmatrix} 1 \\ -4 \\ 3 \end{bmatrix}, B=[−121]B = \begin{bmatrix} -1 & 2 & 1 \end{bmatrix}
  10. Ex 3.3 Q5 (ii)
    A=[012]A = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}, B=[157]B = \begin{bmatrix} 1 & 5 & 7 \end{bmatrix}
  11. Verify the given relation for the following matrix.
    Ex 3.3 Q6 (i)
    If A=[cos⁡αsin⁡α−sin⁡αcos⁡α]A = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix}, then verify that A′A=IA' A = I.
  12. Ex 3.3 Q6 (ii)
    If A=[sin⁡αcos⁡α−cos⁡αsin⁡α]A = \begin{bmatrix} \sin\alpha & \cos\alpha \\ -\cos\alpha & \sin\alpha \end{bmatrix}, then verify that A′A=IA' A = I.
  13. Ex 3.3 Q7 (i)
    Show that the matrix A=[1−15−121513]A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix} is a symmetric matrix.
  14. Ex 3.3 Q7 (ii)
    Show that the matrix A=[01−1−1011−10]A = \begin{bmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix} is a skew symmetric matrix.
  15. For the matrix A=[1567]A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}, verify that
    Ex 3.3 Q8 (i)
    (A+A′)(A + A') is a symmetric matrix
  16. Ex 3.3 Q8 (ii)
    (A−A′)(A - A') is a skew symmetric matrix
  17. Ex 3.3 Q9
    Find 12(A+A′)\frac{1}{2}(A + A') and 12(A−A′)\frac{1}{2}(A - A'), when A=[0ab−a0c−b−c0]A = \begin{bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix}.
  18. Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
    Ex 3.3 Q10 (i)
    [351−1]\begin{bmatrix} 3 & 5 \\ 1 & -1 \end{bmatrix}
  19. Ex 3.3 Q10 (ii)
    [6−22−23−12−13]\begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix}
  20. Ex 3.3 Q10 (iii)
    [33−1−2−21−4−52]\begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix}
  21. Ex 3.3 Q10 (iv)
    [15−12]\begin{bmatrix} 1 & 5 \\ -1 & 2 \end{bmatrix}
  22. Ex 3.3 Q11
    If A, B are symmetric matrices of same order, then AB−BAAB - BA is a
    1. A.
      Skew symmetric matrix
    2. B.
      Symmetric matrix
    3. C.
      Zero matrix
    4. D.
      Identity matrix
  23. Ex 3.3 Q12
    If A=[cos⁡α−sin⁡αsin⁡αcos⁡α]A = \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}, and A+A′=IA + A' = I, then the value of α\alpha is
    1. A.
      π6\frac{\pi}{6}
    2. B.
      π3\frac{\pi}{3}
    3. C.
      π\pi
    4. D.
      3π2\frac{3\pi}{2}

3.7 Elementary Operation (Transformation) of a Matrix

1 q

Exercise 3.4

Practice · 1
  1. Ex 3.4 Q1
    Matrices AA and BB will be inverse of each other only if
    1. A.
      AB=BAAB = BA
    2. B.
      AB=BA=0AB = BA = 0
    3. C.
      AB=0, BA=IAB = 0,\ BA = I
    4. D.
      AB=BA=IAB = BA = I

Miscellaneous Exercise on Chapter 3

15 q

Solved Examples

Worked · 3
  1. Misc Eg.23
    If A=[cos⁡θsin⁡θ−sin⁡θcos⁡θ]A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}, then 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}, n∈Nn \in \mathbf{N}.
  2. Misc Eg.24
    If AA and BB are symmetric matrices of the same order, then show that ABAB is symmetric if and only if AA and BB commute, that is AB=BAAB = BA.
  3. Misc Eg.25
    Let A=[2−134]A = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}, B=[5274]B = \begin{bmatrix} 5 & 2 \\ 7 & 4 \end{bmatrix}, C=[2538]C = \begin{bmatrix} 2 & 5 \\ 3 & 8 \end{bmatrix}. Find a matrix DD such that CD−AB=OCD - AB = O.

Miscellaneous Exercise

Practice · 12
  1. Misc Q1
    If AA and BB are symmetric matrices, prove that AB−BAAB - BA is a skew symmetric matrix.
  2. Misc Q2
    Show that the matrix B′ABB'AB is symmetric or skew symmetric according as AA is symmetric or skew symmetric.
  3. Misc Q3
    Find the values of x,y,zx, y, z if the matrix A=[02yzxy−zx−yz]A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} satisfy the equation A′A=IA'A = I.
  4. Misc Q4
    For what values of xx: [121][120201102][02x]=O\begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix} = O?
  5. Misc Q5
    If A=[31−12]A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}, show that A2−5A+7I=0A^2 - 5A + 7I = 0.
  6. Misc Q6
    Find xx, if [x−5−1][102021203][x41]=O\begin{bmatrix} x & -5 & -1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \begin{bmatrix} x \\ 4 \\ 1 \end{bmatrix} = O.
  7. A manufacturer produces three products x,y,zx, y, z which he sells in two markets. Annual sales are indicated below:
    Marketxxyyzz
    I10,0002,00018,000
    II6,00020,0008,000
    Misc Q7 (a)
    If unit sale prices of x,yx, y and zz are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra.
  8. Misc Q7 (b)
    If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit.
  9. Misc Q8
    Find the matrix XX so that X[123456]=[−7−8−9246]X \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}.
  10. Misc Q9
    If A=[αβγ−α]A = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix} is such that A2=IA^2 = I, then
    1. A.
      1+α2+βγ=01 + \alpha^2 + \beta\gamma = 0
    2. B.
      1−α2+βγ=01 - \alpha^2 + \beta\gamma = 0
    3. C.
      1−α2−βγ=01 - \alpha^2 - \beta\gamma = 0
    4. D.
      1+α2−βγ=01 + \alpha^2 - \beta\gamma = 0
  11. Misc Q10
    If the matrix AA is both symmetric and skew symmetric, then
    1. A.
      AA is a diagonal matrix
    2. B.
      AA is a zero matrix
    3. C.
      AA is a square matrix
    4. D.
      None of these
  12. Misc Q11
    If AA is square matrix such that A2=AA^2 = A, then (I+A)3−7A(I + A)^3 - 7A is equal to
    1. A.
      AA
    2. B.
      I−AI - A
    3. C.
      II
    4. D.
      3A3A