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

12th Standard

    Reg.No. :
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Maths

Time : 00:30:00 Hrs
Total Marks : 45

     3 Marks

    15 x 3 = 45
  1. Solve: 2x + 3y = 10, x + 6y = 4 using Cramer's rule.

  2. For what value of t will the system tx +3y - z = 1, x + 2y + z = 2, -tx + y + 2z = -1 fail to have unique solution?

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

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

  5. Under what conditions will the rank of the matrix \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & h-2 & 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} h+2 \\ 3 \end{matrix} \end{matrix} \right] \) be less than 3?

  6. Find the rank of the matrix math \(\left[ \begin{matrix} 4 \\ -2 \\ 1 \end{matrix}\begin{matrix} 4 \\ 3 \\ 4 \end{matrix}\begin{matrix} 0 \\ -1 \\ 8 \end{matrix}\begin{matrix} 3 \\ 5 \\ 7 \end{matrix} \right] \).

  7. Verify that (A-1)T = (AT)-1 for A =\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).

  8. Solve 2x - 3y = 7, 4x - 6y = 14 by Gaussian Jordan method.

  9. Solve: x + y + 3z = 4, 2x + 2y + 6z = 7, 2x + y +  z = 10.

  10. If the rank of the matrix \(\left[ \begin{matrix} \lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda \end{matrix} \right] \) is 2, then find ⋋.

  11. If A = \(\left[\begin{array}{cc} 1 & -3 \\ -2 & 7 \end{array}\right]\)and B = \(\left[\begin{array}{ll} 7 & 3 \\ 2 & 1 \end{array}\right]\)then show that AB = BA = I and therefore, B = A-1.

  12. Find the inverse, if it exists, of the matrix.
    \(A=\left[\begin{array}{ccc} 0 & 2 & 3 \\ -1 & -3 & 3 \\ 1 & 2 & 2 \end{array}\right]\)

  13. Suppose a matrix A satisfies A2 - 5A + 7I = 0. If A5 = aA + bI then find the value of 2a - 3b.

  14. If A \(=\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos \alpha & -\sin \alpha \\ 0 & \sin \alpha & \cos \alpha \end{array}\right]\), then \(|(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))))|\)

  15. If Ar \(\left|\begin{array}{cc} r & r-1 \\ r-1 & r \end{array}\right|\), where r is a natural number then find the value of \(\sqrt{\left(\sum_{r=1}^{3013} A_{r}\right)}\)

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