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Mathematics · Textbook solutions

Matrices

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 120 questions

2.1 Elementary Transformations

12 q

Solved Examples

Worked · 3
  1. 2.1 Solved Ex.1
    If A=[10−13]A = \begin{bmatrix} 1 & 0 \\ -1 & 3 \end{bmatrix}, apply the transformation R1↔R2R_1 \leftrightarrow R_2 on A.
  2. 2.1 Solved Ex.2
    If A=[102234]A = \begin{bmatrix} 1 & 0 & 2 \\ 2 & 3 & 4 \end{bmatrix}, apply the transformation C1→C1+2C3C_1 \to C_1 + 2C_3.
  3. 2.1 Solved Ex.3
    If A=[12−13−25]A = \begin{bmatrix} 1 & 2 & -1 \\ 3 & -2 & 5 \end{bmatrix}, apply R1↔R2R_1 \leftrightarrow R_2 and then C1→C1+2C3C_1 \to C_1 + 2C_3 on A.

Exercise 2.1

Practice · 9
  1. Apply the given elementary transformation on each of the following matrices.
    Q.1
    A=[10−13]A = \begin{bmatrix} 1 & 0 \\ -1 & 3 \end{bmatrix}, R1↔R2R_1 \leftrightarrow R_2.
  2. Apply the given elementary transformation on each of the following matrices.
    Q.2
    B=[1−13254]B = \begin{bmatrix} 1 & -1 & 3 \\ 2 & 5 & 4 \end{bmatrix}, R1→R1−R2R_1 \to R_1 - R_2.
  3. Apply the given elementary transformation on each of the following matrices.
    Q.3
    A=[5413]A = \begin{bmatrix} 5 & 4 \\ 1 & 3 \end{bmatrix}, C1↔C2C_1 \leftrightarrow C_2; B=[3145]B = \begin{bmatrix} 3 & 1 \\ 4 & 5 \end{bmatrix}, R1↔R2R_1 \leftrightarrow R_2. What do you observe?
  4. Apply the given elementary transformation on each of the following matrices.
    Q.4
    A=[12−1013]A = \begin{bmatrix} 1 & 2 & -1 \\ 0 & 1 & 3 \end{bmatrix}, 2C22C_2; B=[102245]B = \begin{bmatrix} 1 & 0 & 2 \\ 2 & 4 & 5 \end{bmatrix}, −3R1-3R_1. Find the addition of the two new matrices.
  5. Apply the given elementary transformation on each of the following matrices.
    Q.5
    A=[1−13210331]A = \begin{bmatrix} 1 & -1 & 3 \\ 2 & 1 & 0 \\ 3 & 3 & 1 \end{bmatrix}, 3R33R_3 and then C3+2C2C_3 + 2C_2.
  6. Apply the given elementary transformation on each of the following matrices.
    Q.6
    A=[1−13210331]A = \begin{bmatrix} 1 & -1 & 3 \\ 2 & 1 & 0 \\ 3 & 3 & 1 \end{bmatrix}, C3+2C2C_3 + 2C_2 and then 3R33R_3. What do you conclude from ex. 5 and ex. 6?
  7. Q.7
    Use suitable transformation on [1234]\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} to convert it into an upper triangular matrix.
  8. Q.8
    Convert [1−123]\begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix} into an identity matrix by suitable row transformations.
  9. Q.9
    Transform [1−12213324]\begin{bmatrix} 1 & -1 & 2 \\ 2 & 1 & 3 \\ 3 & 2 & 4 \end{bmatrix} into an upper triangular matrix by suitable column transformations.

2.2 Inverse of a Matrix

25 q

Solved Examples

Worked · 10
  1. 2.2 Solved Ex.1
    Find which of the following matrices are invertible: (i) A=[2142]A = \begin{bmatrix} 2 & 1 \\ 4 & 2 \end{bmatrix} (ii) B=[cos⁡θsin⁡θ−sin⁡θcos⁡θ]B = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} (iii) C=[132312123]C = \begin{bmatrix} 1 & 3 & 2 \\ 3 & 1 & 2 \\ 1 & 2 & 3 \end{bmatrix}
  2. 2.2 Solved Ex.2
    Find the inverse of A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}.
  3. 2.2 Solved Ex.3
    Find the inverse of A=[326112225]A = \begin{bmatrix} 3 & 2 & 6 \\ 1 & 1 & 2 \\ 2 & 2 & 5 \end{bmatrix} by using elementary row transformations.
  4. 2.2 Solved Ex.4
    Find the inverse of A=[133143134]A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix} by elementary column transformation.
  5. 2.2 Solved Ex.5
    Find the co-factors of the elements of A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}.
  6. 2.2 Solved Ex.6
    Find the adjoint of matrix A=[2−341]A = \begin{bmatrix} 2 & -3 \\ 4 & 1 \end{bmatrix}.
  7. 2.2 Solved Ex.7
    Find the adjoint of matrix A=[20−1312−112]A = \begin{bmatrix} 2 & 0 & -1 \\ 3 & 1 & 2 \\ -1 & 1 & 2 \end{bmatrix}.
  8. 2.2 Solved Ex.8
    If A=[2−243]A = \begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix}, then find A−1A^{-1} by the adjoint method.
  9. 2.2 Solved Ex.9
    If A=[2−11−12−11−12]A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}, find A−1A^{-1} by the adjoint method.
  10. 2.2 Solved Ex.10
    If A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, verify that A(adj A)=(adj A)A=∣A∣ IA(\text{adj}\,A) = (\text{adj}\,A)A = |A|\,I.

Exercise 2.2

Practice · 15
  1. Find the co-factors of the elements of the following matrices.
    Q.1 (i)
    [−12−34]\begin{bmatrix} -1 & 2 \\ -3 & 4 \end{bmatrix}
  2. Q.1 (ii)
    [1−12−235−20−1]\begin{bmatrix} 1 & -1 & 2 \\ -2 & 3 & 5 \\ -2 & 0 & -1 \end{bmatrix}
  3. Find the matrix of co-factors for the following matrices.
    Q.2 (i)
    [134−1]\begin{bmatrix} 1 & 3 \\ 4 & -1 \end{bmatrix}
  4. Q.2 (ii)
    [102−21303−5]\begin{bmatrix} 1 & 0 & 2 \\ -2 & 1 & 3 \\ 0 & 3 & -5 \end{bmatrix}
  5. Find the adjoint of the following matrices.
    Q.3 (i)
    [2−335]\begin{bmatrix} 2 & -3 \\ 3 & 5 \end{bmatrix}
  6. Q.3 (ii)
    [1−12−235−20−1]\begin{bmatrix} 1 & -1 & 2 \\ -2 & 3 & 5 \\ -2 & 0 & -1 \end{bmatrix}
  7. Q.4
    If A=[1−1230−2103]A = \begin{bmatrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{bmatrix}, verify that A(adj A)=(adj A)A=∣A∣ IA(\text{adj}\,A) = (\text{adj}\,A)A = |A|\,I.
  8. Find the inverse of the following matrices by the adjoint method.
    Q.5 (i)
    [−15−32]\begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix}
  9. Q.5 (ii)
    [2−243]\begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix}
  10. Q.5 (iii)
    [10033052−1]\begin{bmatrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{bmatrix}
  11. Q.5 (iv)
    [123024005]\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{bmatrix}
  12. Find the inverse of the following matrices.
    Q.6 (i)
    [122−1]\begin{bmatrix} 1 & 2 \\ 2 & -1 \end{bmatrix}
  13. Q.6 (ii)
    [2−3−12]\begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}
  14. Q.6 (iii)
    [012123311]\begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{bmatrix}
  15. Q.6 (iv)
    [20−1510013]\begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix}

Miscellaneous Exercise 2 (A)

38 q
  1. Misc 2A Q.1
    If A=[100210331]A = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 3 & 1 \end{bmatrix} then reduce it to I3I_3 by using column transformations.
  2. Misc 2A Q.2
    If A=[213101111]A = \begin{bmatrix} 2 & 1 & 3 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \end{bmatrix} then reduce it to I3I_3 by using row transformations.
  3. Check whether the following matrices are invertible or not.
    Misc 2A Q.3 i)
    Check whether the following matrix is invertible or not: [1001]\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
  4. Misc 2A Q.3 ii)
    Check whether the following matrix is invertible or not: [1111]\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}
  5. Misc 2A Q.3 iii)
    Check whether the following matrix is invertible or not: [1233]\begin{bmatrix} 1 & 2 \\ 3 & 3 \end{bmatrix}
  6. Misc 2A Q.3 iv)
    Check whether the following matrix is invertible or not: [231015]\begin{bmatrix} 2 & 3 \\ 10 & 15 \end{bmatrix}
  7. Misc 2A Q.3 v)
    Check whether the following matrix is invertible or not: [cos⁡θsin⁡θ−sin⁡θcos⁡θ]\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}
  8. Misc 2A Q.3 vi)
    Check whether the following matrix is invertible or not: [sec⁡θtan⁡θtan⁡θsec⁡θ]\begin{bmatrix} \sec\theta & \tan\theta \\ \tan\theta & \sec\theta \end{bmatrix}
  9. Misc 2A Q.3 vii)
    Check whether the following matrix is invertible or not: [343110145]\begin{bmatrix} 3 & 4 & 3 \\ 1 & 1 & 0 \\ 1 & 4 & 5 \end{bmatrix}
  10. Misc 2A Q.3 viii)
    Check whether the following matrix is invertible or not: [1232−13123]\begin{bmatrix} 1 & 2 & 3 \\ 2 & -1 & 3 \\ 1 & 2 & 3 \end{bmatrix}
  11. Misc 2A Q.3 ix)
    Check whether the following matrix is invertible or not: [123345468]\begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 4 & 6 & 8 \end{bmatrix}
  12. Misc 2A Q.4
    Find AB, if A=[1231−2−3]A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & -2 & -3 \end{bmatrix} and B=[1−1121−2]B = \begin{bmatrix} 1 & -1 \\ 1 & 2 \\ 1 & -2 \end{bmatrix}. Examine whether AB has inverse or not.
  13. Misc 2A Q.5
    If A=[x000y000z]A = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix} is a nonsingular matrix then find A−1A^{-1} by elementary row transformations. Hence, find the inverse of [20001000−1]\begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}.
  14. Misc 2A Q.6
    If A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} and X is a 2×22\times2 matrix such that AX=IAX = I, then find X.
  15. Find the inverse of each of the following matrices (if they exist).
    Misc 2A Q.7 i)
    Find the inverse of the following matrix (if it exists): [1−123]\begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}
  16. Misc 2A Q.7 ii)
    Find the inverse of the following matrix (if it exists): [211−1]\begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix}
  17. Misc 2A Q.7 iii)
    Find the inverse of the following matrix (if it exists): [1327]\begin{bmatrix} 1 & 3 \\ 2 & 7 \end{bmatrix}
  18. Misc 2A Q.7 iv)
    Find the inverse of the following matrix (if it exists): [2−357]\begin{bmatrix} 2 & -3 \\ 5 & 7 \end{bmatrix}
  19. Misc 2A Q.7 v)
    Find the inverse of the following matrix (if it exists): [2174]\begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix}
  20. Misc 2A Q.7 vi)
    Find the inverse of the following matrix (if it exists): [3−102−7]\begin{bmatrix} 3 & -10 \\ 2 & -7 \end{bmatrix}
  21. Misc 2A Q.7 vii)
    Find the inverse of the following matrix (if it exists): [2−332233−22]\begin{bmatrix} 2 & -3 & 3 \\ 2 & 2 & 3 \\ 3 & -2 & 2 \end{bmatrix}
  22. Misc 2A Q.7 viii)
    Find the inverse of the following matrix (if it exists): [13−2−30−5250]\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -5 \\ 2 & 5 & 0 \end{bmatrix}
  23. Misc 2A Q.7 ix)
    Find the inverse of the following matrix (if it exists): [20−1510013]\begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix}
  24. Misc 2A Q.7 x)
    Find the inverse of the following matrix (if it exists): [12−20−21−130]\begin{bmatrix} 1 & 2 & -2 \\ 0 & -2 & 1 \\ -1 & 3 & 0 \end{bmatrix}
  25. Misc 2A Q.8
    Find the inverse of A=[cos⁡θ−sin⁡θ0sin⁡θcos⁡θ0001]A = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} by (i) elementary row transformations (ii) elementary column transformations.
  26. Misc 2A Q.9
    If A=[2312]A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}, B=[1031]B = \begin{bmatrix} 1 & 0 \\ 3 & 1 \end{bmatrix} find AB and (AB)−1(AB)^{-1}. Verify that (AB)−1=B−1A−1(AB)^{-1} = B^{-1}A^{-1}.
  27. Misc 2A Q.10
    If A=[4521]A = \begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}, then show that A−1=16(A−5I)A^{-1} = \frac{1}{6}(A - 5I).
  28. Misc 2A Q.11
    Find matrix X such that AX=BAX = B, where A=[12−13]A = \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix} and B=[0124]B = \begin{bmatrix} 0 & 1 \\ 2 & 4 \end{bmatrix}.
  29. Misc 2A Q.12
    Find X, if AX=BAX = B where A=[123−112124]A = \begin{bmatrix} 1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4 \end{bmatrix} and B=[123]B = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}.
  30. Misc 2A Q.13
    If A=[1112]A = \begin{bmatrix} 1 & 1 \\ 1 & 2 \end{bmatrix}, B=[4131]B = \begin{bmatrix} 4 & 1 \\ 3 & 1 \end{bmatrix} and C=[247319]C = \begin{bmatrix} 24 & 7 \\ 31 & 9 \end{bmatrix} then find matrix X such that AXB=CAXB = C.
  31. Misc 2A Q.14
    Find the inverse of [123115247]\begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 5 \\ 2 & 4 & 7 \end{bmatrix} by adjoint method.
  32. Misc 2A Q.15
    Find the inverse of [101023121]\begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1 \end{bmatrix} by adjoint method.
  33. Misc 2A Q.16
    Find A−1A^{-1} by adjoint method and by elementary transformations if A=[123−112124]A = \begin{bmatrix} 1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4 \end{bmatrix}.
  34. Misc 2A Q.17
    Find the inverse of A=[101023121]A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1 \end{bmatrix} by elementary column transformations.
  35. Misc 2A Q.18
    Find the inverse of [123115247]\begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 5 \\ 2 & 4 & 7 \end{bmatrix} by elementary row transformations.
  36. Show with usual notations that for any matrix A=[aij]3×3A = [a_{ij}]_{3 \times 3}.
    Misc 2A Q.19 i)
    Show with usual notations that for any matrix A=[aij]3×3A = [a_{ij}]_{3 \times 3}: a11A21+a12A22+a13A23=0a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} = 0
  37. Misc 2A Q.19 ii)
    Show with usual notations that for any matrix A=[aij]3×3A = [a_{ij}]_{3 \times 3}: a11A11+a12A12+a13A13=∣A∣a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13} = |A|
  38. Misc 2A Q.20
    If A=[101023121]A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1 \end{bmatrix} and B=[123115247]B = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 5 \\ 2 & 4 & 7 \end{bmatrix} then, find a matrix X such that XA=BXA = B.

2.3 Application of Matrices

16 q

Solved Examples

Worked · 6
  1. (inversion)
    Solve the equations 2x+5y=12x + 5y = 1 and 3x+2y=73x + 2y = 7 by the method of inversion.
  2. (inversion)
    Solve the following equations by the method of inversion: x−y+z=4, 2x+y−3z=0, x+y+z=2x - y + z = 4,\ 2x + y - 3z = 0,\ x + y + z = 2.
  3. (reduction)
    Solve the equations 2x+3y=92x + 3y = 9 and y−x=−2y - x = -2 using the method of reduction.
  4. (reduction)
    Solve the following equations by the method of reduction: x+3y+3z=12, x+4y+4z=15x + 3y + 3z = 12,\ x + 4y + 4z = 15 and x+3y+4z=13x + 3y + 4z = 13.
  5. (reduction)
    Solve the following equations by the method of reduction: x+y+z=1, 2x+3y+2z=2x + y + z = 1,\ 2x + 3y + 2z = 2 and x+y+2z=4x + y + 2z = 4.
  6. (reduction)
    The cost of 2 books and 6 note books is Rs. 34 and the cost of 3 books and 4 notebooks is Rs. 31. Using matrices, find the cost of one book and one note-book.

Exercise 2.3

Practice · 10
  1. Solve the following equations by inversion method.
    Q.1 (i)
    Solve by inversion method: x+2y=2, 2x+3y=3x + 2y = 2,\ 2x + 3y = 3.
  2. Q.1 (ii)
    Solve by inversion method: x+y=4, 2x−y=5x + y = 4,\ 2x - y = 5.
  3. Q.1 (iii)
    Solve by inversion method: 2x+6y=8, x+3y=52x + 6y = 8,\ x + 3y = 5.
  4. Solve the following equations by reduction method.
    Q.2 (i)
    Solve by reduction method: 2x+y=5, 3x+5y=−32x + y = 5,\ 3x + 5y = -3.
  5. Q.2 (ii)
    Solve by reduction method: x+3y=2, 3x+5y=4x + 3y = 2,\ 3x + 5y = 4.
  6. Q.2 (iii)
    Solve by reduction method: 3x−y=1, 4x+y=63x - y = 1,\ 4x + y = 6.
  7. Q.2 (iv)
    Solve by reduction method: 5x+2y=4, 7x+3y=55x + 2y = 4,\ 7x + 3y = 5.
  8. Q.3
    The cost of 4 pencils, 3 pens and 2 erasers is Rs. 60. The cost of 2 pencils, 4 pens and 6 erasers is Rs. 90, whereas the cost of 6 pencils, 2 pens and 3 erasers is Rs. 70. Find the cost of each item by using matrices.
  9. Q.4
    If three numbers are added, their sum is 2. If 2 times the second number is subtracted from the sum of first and third number we get 8, and if three times the first number is added to the sum of second and third number we get 4. Find the numbers using matrices.
  10. Q.5
    The total cost of 3 T.V. sets and 2 V.C.R.s is Rs. 35000. The shop-keeper wants profit of Rs. 1000 per television and Rs. 500 per V.C.R. He can sell 2 T.V. sets and 1 V.C.R. and get the total revenue as Rs. 21,500. Find the cost price and the selling price of a T.V. set and a V.C.R.

Miscellaneous Exercise 2 (B)

29 q

Choose the correct alternative

Practice · 12
  1. Misc I (1)
    If A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, adj(A)=[4a−3b]\text{adj}(A) = \begin{bmatrix} 4 & a \\ -3 & b \end{bmatrix} then the values of a and b are,

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  2. Misc I (2)
    The inverse of [0110]\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} is

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  3. Misc I (3)
    If A=[1221]A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} and A(adjA)=kIA(\text{adj}A) = kI then the value of k is .....

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  4. Misc I (4)
    If A=[2−431]A = \begin{bmatrix} 2 & -4 \\ 3 & 1 \end{bmatrix} then the adjoint of matrix A is

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  5. Misc I (5)
    If A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} and A(adj A)=kIA(\text{adj }A) = kI then the value of k is

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  6. Misc I (6)
    If A=[λ1−1−λ]A = \begin{bmatrix} \lambda & 1 \\ -1 & -\lambda \end{bmatrix} then A−1A^{-1} does not exist if λ=\lambda =

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  7. Misc I (7)
    If A=[cos⁡α−sin⁡αsin⁡αcos⁡α]A = \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix} then A−1=A^{-1} =

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  8. Misc I (8)
    If F(α)=[cos⁡α−sin⁡α0sin⁡αcos⁡α0001]F(\alpha) = \begin{bmatrix} \cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1 \end{bmatrix} where α∈R\alpha \in R then [F(α)]−1[F(\alpha)]^{-1} is =

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  9. Misc I (9)
    The inverse of A=[010100001]A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} is

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  10. Misc I (10)
    The inverse of a symmetric matrix is -

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  11. Misc I (11)
    For a 2×22 \times 2 matrix A, if A(adjA)=[100010]A(\text{adj}A) = \begin{bmatrix} 10 & 0 \\ 0 & 10 \end{bmatrix} then determinant A equals

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  12. Misc I (12)
    If A−1=−12[1−4−12]A^{-1} = -\dfrac{1}{2}\begin{bmatrix} 1 & -4 \\ -1 & 2 \end{bmatrix} then A=A =

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Answer the following

Practice · 17
  1. Solve the following equations by the methods of inversion.
    Misc II Q.1 (i)
    2x−y=−2, x+5y=52x - y = -2,\ x + 5y = 5
  2. Misc II Q.1 (ii)
    x+y+z=1, 2x+3y+2z=2x + y + z = 1,\ 2x + 3y + 2z = 2 and ax+ay+2az=4ax + ay + 2az = 4
  3. Misc II Q.1 (iii)
    5x−y+4z=5, 2x+3y+5z=25x - y + 4z = 5,\ 2x + 3y + 5z = 2 and 5x−2y+6z=−15x - 2y + 6z = -1
  4. Misc II Q.1 (iv)
    2x+3y=−5, 3x+y=32x + 3y = -5,\ 3x + y = 3
  5. Misc II Q.1 (v)
    x+y+z=−1, y+z=2x + y + z = -1,\ y + z = 2 and x+y−z=3x + y - z = 3
  6. Express the following equation in matrix form and solve them by the method of reduction.
    Misc II Q.2 (i)
    x−y+z=1, 2x−y=1, 3x+3y−4z=2x - y + z = 1,\ 2x - y = 1,\ 3x + 3y - 4z = 2
  7. Misc II Q.2 (ii)
    x+y=1, y+z=53, z+x=43x + y = 1,\ y + z = \dfrac{5}{3},\ z + x = \dfrac{4}{3}
  8. Misc II Q.2 (iii)
    2x−y+z=1, x+2y+3z=82x - y + z = 1,\ x + 2y + 3z = 8 and 3x+y−4z=13x + y - 4z = 1
  9. Misc II Q.2 (iv)
    x+y+z=6, 3x−y+3z=10x + y + z = 6,\ 3x - y + 3z = 10 and 5x+5y−4z=35x + 5y - 4z = 3
  10. Misc II Q.2 (v)
    x+2y+z=8, 2x+3y−z=1x + 2y + z = 8,\ 2x + 3y - z = 1 and 3x−y−2z=53x - y - 2z = 5
  11. Misc II Q.2 (vi)
    x+3y+2z=6, 3x−2y+5z=5x + 3y + 2z = 6,\ 3x - 2y + 5z = 5 and 2x−3y+6z=72x - 3y + 6z = 7
  12. Misc II Q.3
    The sum of three numbers is 6. If we multiply third number by 3 and add it to the second number we get 11. By adding first and the third numbers we get a number which is double the second number. Use this information and find a system of linear equations. Find the three numbers using matrices.
  13. Misc II Q.4
    The cost of 4 pencils, 3 pens and 2 books is Rs.150. The cost of 1 pencil, 2 pens and 3 books is Rs.125. The cost of 6 pencils, 2 pens and 3 books is Rs.175. Find the cost of each item by using Matrices.
  14. Misc II Q.5
    The sum of three numbers is 6. Thrice the third number when added to the first number gives 7. On adding three times first number to the sum of second and third number we get 12. Find the three numbers by using Matrices.
  15. Misc II Q.6
    The sum of three numbers is 2. If twice the second number is added to the sum of first and third number, we get 1. On adding five times the first number to the sum of second and third we get 6. Find the three numbers by using matrices.
  16. Misc II Q.7
    An amount of Rs.5000 is invested in three types of investments, at interest rates 6%, 7%, 8% per annum respectively. The total annual income from these investments is Rs.350. If the total annual income from first two investments is Rs.70 more than the income from the third, find the amount of each investment using matrix method.
  17. Misc II Q.8
    The sum of the costs of one book each of Mathematics, Physics and Chemistry is Rs.210. Total cost of a mathematics book, 2 physics books, and a chemistry book is Rs.240. Also the total cost of a Mathematics book, 3 physics books and 2 chemistry books is Rs.300. Find the cost of each book, using Matrices.