#### Application of Matrices and Determinants Important Questions

12th Standard EM

Reg.No. :
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Mathematics

Time : 01:00:00 Hrs
Total Marks : 50

Part - A

10 x 1 = 10
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. If A = $\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right]$, B = adj A and C = 3A, then $\frac { \left| adjB \right| }{ \left| C \right| }$

(a)

$\frac { 1 }{ 3 }$

(b)

$\frac { 1 }{ 9 }$

(c)

$\frac { 1 }{ 4 }$

(d)

1

4. If A$\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right]$, then A =

(a)

$\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right]$

(b)

$\left[ \begin{matrix} 1 & 2 \\ -1 & 4 \end{matrix} \right]$

(c)

$\left[ \begin{matrix} 4 & 2 \\ -1 & 1 \end{matrix} \right]$

(d)

$\left[ \begin{matrix} 4 & -1 \\ 2 & 1 \end{matrix} \right]$

5. If A = $\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right]$, then 9I - A =

(a)

A-1

(b)

$\frac { { A }^{ -1 } }{ 2 }$

(c)

3A-1

(d)

2A-1

6. 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

7. Let A be a 3 x 3 matrix and B its adjoint matrix If |B|=64, then |A|=

(a)

±2

(b)

±4

(c)

±8

(d)

±12

8. If AT is the transpose of a square matrix A, then

(a)

|A| ≠ |AT|

(b)

|A| = |AT|

(c)

|A| + |AT| =0

(d)

|A| = |AT| only

9. The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is

(a)

0

(b)

1

(c)

2

(d)

infinitely many

10. If A is a square matrix that IAI = 2, than for any positive integer n, |An| =

(a)

0

(b)

2n

(c)

2n

(d)

n2

11. Part -B

5 x 2 = 10
12. 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(-$\alpha$).

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

14. 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.

15. For any 2 x 2 matrix, if A (adj A) =$\left[ \begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix} \right]$ then find |A|.

16. For the matrix A, if A3 = I, then find A-1.

17. Part - C

5 x 3 = 15
18. Find the inverse of the matrix $\left[ \begin{matrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \end{matrix} \right]$.

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

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

21. Under what co.nditions 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?

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

23. Part - D

3 x 5 = 15
24. 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.

25. Solve: $\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z }$=2

26. Using Gaussian Jordan method, find the values of λ and μ so that the system of equations 2x - 3y + 5z = 12, 3x + y + λz =μ, x - 7y + 8z = 17 has (i) unique solution (ii) infinite solutions and (iii) no solution.