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#### Application of Matrices and Determinants Book Back Questions

12th Standard EM

Reg.No. :
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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 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 = $\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right]$, then 9I - A =

(a)

A-1

(b)

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

(c)

3A-1

(d)

2A-1

5. If A = $\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right]$ and B = $\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right]$ then |adj (AB)| =

(a)

-40

(b)

-80

(c)

-60

(d)

-20

6. 5 x 2 = 10
7. If A = $\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right]$ is non-singular, find A−1.

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

9. If adj A = $\left[ \begin{matrix} -1 & 2 & 2 \\ 1 & 1 & 2 \\ 2 & 2 & 1 \end{matrix} \right]$, find A−1.

10. If A is symmetric, prove that then adj Ais also symmetric.

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

12. 5 x 3 = 15
13. Find the rank of the following matrices by row reduction method:
$\left[ \begin{matrix} 1 \\ \begin{matrix} 3 \\ \begin{matrix} 1 \\ 1 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} -1 \\ \begin{matrix} -2 \\ -1 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} -1 \\ \begin{matrix} 2 \\ \begin{matrix} 3 \\ 1 \end{matrix} \end{matrix} \end{matrix} \right]$

14. Find the rank of the following matrices by row reduction method:
$\left[ \begin{matrix} 3 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} -8 \\ \begin{matrix} -5 \\ 2 \end{matrix} \end{matrix}\begin{matrix} 5 \\ \begin{matrix} 1 \\ 3 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix} \right]$

15. Solve the following system of linear equations by matrix inversion method:
2x − y = 8, 3x + 2y = −2

16. Solve the following systems of linear equations by Cramer’s rule:
$\frac { 3 }{ x }$ + 2y = 12, $\frac { 2 }{ x }$ + 3y = 13

17. Find the rank of each of the following matrices:
$\left[ \begin{matrix} 4 & 3 \\ -3 & -1 \\ 6 & 7 \end{matrix}\begin{matrix} 1 & -2 \\ -2 & 4 \\ -1 & 2 \end{matrix} \right]$

18. 4 x 5 = 20
19. Solve the following system of linear equations by matrix inversion method:
2x + 3y − z = 9, x + y + z = 9, 3x − y − z  = −1

20. Solve the following system of linear equations by matrix inversion method:
x + y + z − 2 = 0, 6x − 4y + 5z − 31 = 0, 5x + 2y + 2z = 13.

21. Solve the following systems of linear equations by Cramer’s rule:
3x + 3y − z = 11, 2x − y + 2z = 9, 4x + 3y + 2z = 25.

22. Solve the following systems of linear equations by Cramer’s rule:
$\frac { 3 }{ x } -\frac { 4 }{ y } -\frac { 2 }{ z }$ - 0, $\frac { 1 }{ x } +\frac { 2 }{ y } +\frac { 1 }{ z }$ - 2 = 0, $\frac { 2 }{ x } -\frac { 5 }{ y } -\frac { 4 }{ z }$ + 1 = 0