New ! Maths MCQ Practise Tests



Application of Matrices and Determinants 3 Mark Book Back Question Paper With Answer Key

12th Standard

    Reg.No. :
  •  
  •  
  •  
  •  
  •  
  •  

Maths

Time : 01:00:00 Hrs
Total Marks : 189

     3 Marks 

    63 x 3 = 189
  1. If A = \(\left[ \begin{matrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{matrix} \right] \), verify thatA(adj A) = (adj A)A = |A| I3.

  2. Find the inverse of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \end{matrix} \right] \).

  3. Find a matrix A if adj(A) = \(\left[ \begin{matrix} 7 & 7 & -7 \\ -1 & 11 & 7 \\ 11 & 5 & 7 \end{matrix} \right] \).

  4. If adj A = \(\left[ \begin{matrix} -1 & 2 & 2 \\ 1 & 1 & 2 \\ 2 & 2 & 1 \end{matrix} \right] \), find A−1.

  5. Verify the property (AT)-1 = (A-1)T with A = \(\left[ \begin{matrix} 2 & 9 \\ 1 & 7 \end{matrix} \right] \).

  6. Find the inverse (if it exists) of the following:
    \(\left[ \begin{matrix} -2 & 4 \\ 1 & -3 \end{matrix} \right] \)

  7. If F(\(\alpha\)) = \(\left[ \begin{matrix} \cos { \alpha } & 0 & \sin { \alpha } \\ 0 & 1 & 0 \\ -\sin { \alpha } & 0 & \cos { \alpha } \end{matrix} \right] \), show that [F(\(\alpha\))]-1 = F(-\(\alpha\)).

  8. If A = \(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] \), show that A2 - 3A - 7I2 = O2. Hence find A−1.

  9. If A = \(\frac { 1 }{ 9 } \left[ \begin{matrix} -8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & -8 & 4 \end{matrix} \right] \), prove that A−1 = AT.

  10. If A = \(\left[ \begin{matrix} 8 & -4 \\ -5 & 3 \end{matrix} \right] \), verify that A(adj A) = (adj A)A = |A|I2.

  11. If A = \(\left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} -1 & -3 \\ 5 & 2 \end{matrix} \right] \), verify that (AB)-1 = B-1A-1

  12. Decrypt the received encoded message \(\left[ \begin{matrix} 2 & -3 \end{matrix} \right] \left[ \begin{matrix} 20 & 4 \end{matrix} \right] \) with the encryption matrix \(\left[ \begin{matrix} -1 & -1 \\ 2 & 1 \end{matrix} \right] \) and the decryption matrix as its inverse, where the system of codes are described by the numbers 1 - 26 to the letters A - Z respectively, and the number 0 to a blank space.

  13. Reduce the matrix \(\left[ \begin{matrix} 3 & -1 & 2 \\ -6 & 2 & 4 \\ -3 & 1 & 2 \end{matrix} \right] \) to a row-echelon form.

  14. Reduce the matrix \(\left[ \begin{matrix} 0 \\ -1 \\ 4 \end{matrix}\begin{matrix} 3 \\ 0 \\ 2 \end{matrix}\begin{matrix} 1 \\ 2 \\ 0 \end{matrix}\begin{matrix} 6 \\ 5 \\ 0 \end{matrix} \right] \) to row-echelon form.

  15. Find the rank of the matrix \(\left[ \begin{matrix} 2 \\ \begin{matrix} -3 \\ 6 \end{matrix} \end{matrix}\begin{matrix} -2 \\ \begin{matrix} 4 \\ 2 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} -2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} -1 \\ 7 \end{matrix} \end{matrix} \right] \) by reducing it to an echelon form.

  16. Show that the matrix \(\left[ \begin{matrix} 3 & 1 & 4 \\ 2 & 0 & -1 \\ 5 & 2 & 1 \end{matrix} \right] \) is non-singular and reduce it to the identity matrix by elementary row transformations.

  17. Find the inverse of A = \(\left[ \begin{matrix} 2 & 1 & 1 \\ 3 & 2 & 1 \\ 2 & 1 & 2 \end{matrix} \right] \) by Gauss-Jordan method.

  18. Find the rank of the following matrices by row reduction method:
    \(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ 5 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} -1 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 3 \\ 7 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 4 \\ 11 \end{matrix} \end{matrix} \right] \) 

  19. Solve the following system of equations, using matrix inversion method:
    2x1 + 3x2 + 3x3 = 5, x1 - 2x2 + x3 = -4, 3x1 - x2 - 2x3 = 3.

  20. Solve the following systems of linear equations by Cramer’s rule:
    5x − 2y +16 = 0, x + 3y − 7 = 0

  21. Solve the following systems of linear equations by Gaussian elimination method:
    2x − 2y + 3z = 2, x + 2y − z = 3, 3x − y + 2z = 1

  22. Test the consistency of the following system of linear equations
    x - y + z = -9, 2x - y + z = 4, 3x - y + z = 6, 4x - y + 2z = 7.

  23. Find the condition on a, b and c so that the following system of linear equations has one parameter family of solutions: x + y + z = a, x + 2y + 3z = b, 3x + 5y + 7z = c.

  24. Investigate for what values of λ and μ the system of linear equations x  +  2y  +  z  =  7 ,   x  +  y  +  λz   =  μ ,   x  +  3y  −  5z   =  5 has
    (i) no solution 
    (ii) a unique solution 
    (iii) an infinite number of solutions

  25. Test for consistency and if possible, solve the following systems of equations by rank method.
    2x - y + z = 2, 6x - 3y + 3z = 6, 4x - 2y + 2z = 4

  26. Find the value of k for which the equations
    kx - 2y + z = 1, x - 2ky + z = -2, x - 2y + kz = 1 have
    (i) no solution
    (ii) unique solution
    (iii) infinitely many solution

  27. Investigate the values of λ and μ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y - 5z = 8, 2x + 3y + λz = μ, have
    (i) no solution
    (ii) a unique solution
    (iii) an infinite number of solutions.

  28. Solve the following system:
    x + 2y + 3z = 0, 3x + 4y + 4z = 0, 7x + 10y + 12z = 0.

  29. Solve the system: x + 3y - 2z = 0, 2x - y + 4z = 0, x - 11y + 14z = 0

  30. Solve the system: x + y − 2z = 0, 2x − 3y + z = 0, 3x − 7y + 10z = 0, 6x − 9y + 10z = 0.

  31. Determine the values of λ for which the following system of equations (3λ − 8)x + 3y + 3z = 0, 3x + (3λ − 8)y + 3z = 0, 3x + 3y + (3λ − 8)z = 0. has a non-trivial solution.

  32. Solve the following system of homogenous equations.
    3x + 2y + 7z = 0, 4x − 3y − 2z = 0, 5x + 9y + 23z = 0

  33. Find the inverse (if it exists) of the following:
    \(\left[ \begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{matrix} \right] \)

  34. Find the inverse (if it exists) of the following:
    \(\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right] \)

  35. Find the rank of the following matrices by minor method:
    \(\left[ \begin{matrix} 1 & -2 & 3 \\ 2 & 4 & -6 \\ 5 & 1 & -1 \end{matrix} \right] \)

  36. Find the rank of the following matrices by minor method or show that the rank of matrix is 3
    \(\left[ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 8 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 2 \\ 1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ 0 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 3 \\ 2 \end{matrix} \end{matrix} \right] \)

  37. Find the rank of the following matrices by row reduction method:
    \(\left[ \begin{matrix} 1 \\ \begin{matrix} 3 \\ \begin{matrix} 1 \\ 1 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} -1 \\ \begin{matrix} -2 \\ -1 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} -1 \\ \begin{matrix} 2 \\ \begin{matrix} 3 \\ 1 \end{matrix} \end{matrix} \end{matrix} \right] \)

  38. Find the rank of the following matrices by row reduction method:
    \(\left[ \begin{matrix} 3 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} -8 \\ \begin{matrix} -5 \\ 2 \end{matrix} \end{matrix}\begin{matrix} 5 \\ \begin{matrix} 1 \\ 3 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix} \right] \)

  39. Find the inverse of each of the following by Gauss-Jordan method:
    \(\left[ \begin{matrix} 1 & -1 & 0 \\ 1 & 0 & -1 \\ 6 & -2 & -3 \end{matrix} \right] \)

  40. Find the inverse of each of the following by Gauss-Jordan method:
    \(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8 \end{matrix} \right] \)

  41. Solve the following system of linear equations by matrix inversion method:
    2x + 3y − z = 9, x + y + z = 9, 3x − y − z  = −1

  42. Solve the following system of linear equations by matrix inversion method:
    x + y + z − 2 = 0, 6x − 4y + 5z − 31 = 0, 5x + 2y + 2z = 13.

  43. Solve the following systems of linear equations by Cramer’s rule:
     \(\frac { 3 }{ x } \) + 2y = 12, \(\frac { 2 }{ x } \) + 3y = 13

  44. Solve the following system of homogenous equations.
    2x + 3y − z = 0, x − y − 2z = 0, 3x + y + 3z = 0

  45. Find the rank of each of the following matrices:
    \(\left[ \begin{matrix} 4 & 3 \\ -3 & -1 \\ 6 & 7 \end{matrix}\begin{matrix} 1 & -2 \\ -2 & 4 \\ -1 & 2 \end{matrix} \right] \)

  46. For every square matrix A of order n , \(A(\operatorname{adj} A)=(\operatorname{adj} A)|A=| A \mid I_{n}\)

  47. If a square matrix has an inverse, then it is unique

  48. Let A be square matrix of order n. Then, A−1 exists if and only if A is non-singular.

  49. If A is non-singular, then 
    \((i)\ \left|A^{-1}\right|=\frac{1}{|A|} \)
    \((ii)\ \left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T} \)
    \((iii)\ (\lambda A)^{-1}=\frac{1}{\lambda} A^{-1},\)
    where is \(\lambda\) non-zero scalar

  50. Let A, B, and C be square matrices of order n. If A is non-singular and AB = AC, then B = C. (Left Cancellation Law)

  51. Let A, B, and C be square matrices of order n. If A is non-singular and BA = CA, then B = C. (Right Cancellation Law)

  52. If A and B are non-singular matrices of the same order, then the product AB is also non singular and (AB)−1 = B−1A−1. (Reversal Law for Inverses)

  53. If A is non-singular, then A−1 is also non-singular and (A−1)−1 = A.(Law of Double Inverse)

  54. If A is a non-singular square matrix of order n, then
    \((i) (\operatorname{adj} A)^{-1}=\operatorname{adj}\left(A^{-1}\right)=\frac{1}{|A|} A \)
    \((ii) |\operatorname{adj} A|=|A|^{n-1}\)
    \({(iii) } \operatorname{adj}(\operatorname{adj} A)=|A|^{n-2} A\)
    \( (iv) \operatorname{adj}(\lambda A)=\lambda^{n-1} \operatorname{adj}(A), \lambda\)
     is a non zero scalar
    \((v) |\operatorname{adj}(\operatorname{adj} A)|=|A|^{(n-1)^{2}}\)
    \((vi) (\operatorname{adj} A)^{T}=\operatorname{adj}\left(A^{T}\right)\)

  55. If A and B are any two non-singular square matrices of order n , then adj(AB) = (adj B)(adj A).

  56. The rank of a matrix in row echelon form is the number of non-zero rows in it.

  57. The rank of a non-zero matrix is equal to the number of non-zero rows in a row-echelon form of the matrix

  58. Every non-singular matrix can be transformed to an identity matrix, by a sequence of elementary row operations.

  59. show that the distance from the origin to the plane 3x + 6y + 2z + 7 = 0 is 1

  60. Test for consistency and if possible, solve the following systems of equations by rank method or solve the system of equations by cramer's rule.
    x - y + 2z = 2, 2x + y + 4z = 7, 4x - y + z = 4

  61. Test for consistency and if possible, solve the following systems of equations by rank method
    3x + y + z = 2, x - 3y + 2z = 1, 7x - y + 4z = 5

  62. Test for consistency and if possible, solve the following systems of equations by rank method
    2x + 2y + z = 5, x - y + z = 1, 3x + y + 2z = 4

  63. Solve the following systems of linear equations by Gaussian elimination method:
    2x + 4y + 6z = 22, 3x + 8y + 5z = 27, −x + y + 2z = 2

*****************************************

Reviews & Comments about 12th Standard Maths English Medium Application of Matrices and Determinants 3 Mark Book Back Question Paper and Answer Key 2022 - 2023

Write your Comment