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#### Application of Matrices and Determinants Model Question Paper

12th Standard EM

Reg.No. :
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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
(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.