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Application of Matrices and Determinants Model Question Paper

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 35
    4 x 1 = 4
  1. If |adj(adj A)| = |A|9, then the order of the square matrix A is

    (a)

    3

    (b)

    4

    (c)

    2

    (d)

    5

  2. If A is a 3 × 3 non-singular matrix such that AAT = ATA and B = A-1AT, then BBT = 

    (a)

    A

    (b)

    B

    (c)

    I

    (d)

    BT

  3. The system of linear equations x + y + z  = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if

    (a)

    λ = 8

    (b)

    λ = 8, μ ≠ 36

    (c)

    λ ≠ 8

    (d)

    none

  4. If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then

    (a)

    a2 + b2 + c2 = 1

    (b)

    abc ≠ 1

    (c)

    a + b + c =0

    (d)

    a2 + b2 + c2 + 2abc =1

  5. 2 x 2 = 4
  6. The rank of any 3 x 4 matrix is
    (1) May be 1
    (2) May be 2
    (3) May be 3
    (4) Maybe 4

  7. If A is symmetric then
    (1) AT= A
    (2) adj A is symmetric
    (3) adj (AT)= (adj A)T
    (4) A is orthogonal

  8. 5 x 2 = 10
  9. If A = \(\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \) is non-singular, find A−1.

  10. If A is a non-singular matrix of odd order, prove that |adj A| is positive

  11. Find the rank of the matrix \(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{matrix} \right] \) by reducing it to a row-echelon form.

  12. Flod the rank of the matrix \(\left[ \begin{matrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{matrix} \right] \).

  13. Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).

  14. 4 x 3 = 12
  15. Find a matrix A if adj(A) = \(\left[ \begin{matrix} 7 & 7 & -7 \\ -1 & 11 & 7 \\ 11 & 5 & 7 \end{matrix} \right] \).

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

  17. Solve: 3x+ay =4, 2x+ay=2, a≠0 by Cramer's rule.

  18. Verify (AB)-1 =B-1 A-1 for A=\(\left[ \begin{matrix} 2 & 1 \\ 5 & 3 \end{matrix} \right] \) and B=\(\left[ \begin{matrix} 4 & 5 \\ 3 & 4 \end{matrix} \right] \).

  19. 1 x 5 = 5
  20. 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.

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