" /> -->

#### Application of Matrices and Determinants Model Question Paper 1

12th Standard EM

Reg.No. :
•
•
•
•
•
•

Maths

Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
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$\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right]$, 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)

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

3. If A = $\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right]$ and A-1 = $\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right]$ then the value of a23 is

(a)

0

(b)

-2

(c)

-3

(d)

-1

4. 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

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

(a)

0

(b)

2n

(c)

2n

(d)

n2

6. 5 x 2 = 10
7. If A is symmetric, prove that then adj Ais also symmetric.

8. Prove that $\left[ \begin{matrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{matrix} \right]$ is orthogonal

9. Show that the equations 3x + y + 9z = 0, 3x + 2y + 12z = 0 and 2x + y + 7z = 0 have nontrivial solutions also.

10. Solve : 2x - y = 3, 5x + y = 4 using matrices.

11. Solve 6x - 7y = 16, 9x - 5y = 35 using (Cramer's rule).

12. 5 x 3 = 15
13. Verify (AB)-1 = B-1A-1 with A = $\left[ \begin{matrix} 0 & -3 \\ 1 & 4 \end{matrix} \right]$, B = $\left[ \begin{matrix} -2 & -3 \\ 0 & -1 \end{matrix} \right]$.

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

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

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

18. 4 x 5 = 20
19. 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(-$\alpha$).

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

21. Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c =0.

22. 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.