# 87. Matrices Problems Part 2

1. If $$A = \begin{bmatrix}1 & 2 & 3 \\ 2 & 3 & 2 \\ 3 & 3 & 4\end{bmatrix},$$ find $$adj(A)$$

2. For the matrix $$\begin{bmatrix}\cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{bmatrix}$$ verify that $$A (adj A) = |A|I$$

3. For the matrix $$A = \begin{bmatrix}1 & -1 & 1 \\ 2 & 3 & 0 \\ 8 & 2 & 10\end{bmatrix},$$ show that $$A adj(A) = O$$

4. Find the inverse of $$\begin{bmatrix}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{bmatrix}$$

5. Find the inverse of $$\begin{bmatrix}2 & -3 & 3 \\ 2 & 2 & 3 \\ 3 & -2 & 2\end{bmatrix}$$

6. Find the inverse of $$\begin{bmatrix}1 & -2 & 3 \\ 0 & -1 & 4 \\ -2 & 2 & 1\end{bmatrix}$$

7. Find the inverse of $$\begin{bmatrix}1 & 2 & 3 \\ -3 & 5 & 0 \\ 0 & 1 & 1\end{bmatrix}$$

8. If $$A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$$ such that $$ad - bc \neq 0,$$ then find the inverse of $$A.$$

9. If $$A = \begin{bmatrix}3 & 1 \\ 4 & 0\end{bmatrix}, B = \begin{bmatrix} 4 & 0 \\ 2 & 5 \end{bmatrix},$$ verify thet $$(AB)^{-1} = B^{-1}A^{-1}$$

10. If $$A = \begin{bmatrix}1 & \tan x \\ -\tan x & 1\end{bmatrix},$$ show that $$AA^{-1} = \begin{bmatrix}\cos 2x & -\sin 2x \\ \sin 2x & \cos 2x\end{bmatrix}$$

11. If $$A = \begin{bmatrix} 3 & 2 \\ 7 & 5 \end{bmatrix}$$ and $$B = \begin{bmatrix} 6 & 7 \\ 8 & 9 \end{bmatrix},$$ find $$(AB)^{-1}$$

12. Solve the following system of equations by matrix method:

$$\begin{array}{rr}3x - 2y & = 7\\ 5x + 3y & = 1\end{array}$$

13. Solve the following system of equations by matrix method:

$$\begin{array}{rr}5x - 7y & = 2 \\ 7x -5y & = 3\end{array}$$

14. Solve the following system of equations by matrix method:

$$\begin{array}{rr}2x - 3y + 3z & = 1 \\ 2x + 2y + 3z & = 2\\3x -2y + 2z & = 3\end{array}$$

15. Solve the following system of equations by matrix method:

$$\begin{array}{rr}x + y + z & = 3 \\ 2x - y + z & = 2 \\ x - 2y + 3z & = 2\end{array}$$

16. Solve the following system of equations by matrix method:

$$\begin{array}{rr}2x - y + 3z & = 9 \\ x + y + z & = 6 \\ x - y + z & = 2\end{array}$$

17. Examine following system of equations for consistency:

$$\begin{array}{rr} 2x + 3y & = 5 \\ 6x + 9y & = 10\end{array}$$

18. Examine following system of equations for consistency:

$$\begin{array}{rr} 4x - 2y & = 3 \\ 6x - 3y & = 5\end{array}$$