Find the values of \( x, y, z \) if the matrix \( A \) satisfies the equation \( A^T A = I \), where
\[ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} \]
If matrix \[ A = \begin{bmatrix} 1 & 1 & 3 \\ 1 & 3 & -3 \\ -2 & -4 & -4 \end{bmatrix}, \] then find \( A^{-1} \).
If \[ A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} \] prove that \[ A^n = \begin{bmatrix} \cos n\theta & \sin n\theta \\ -\sin n\theta & \cos n\theta \end{bmatrix}, \] where \( n \in \mathbb{N} \).
Find matrix \( AB \) if
\[ A = \begin{bmatrix} -1 & 2 & 3 \\ 4 & -2 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 1 \\ 6 & 7 \\ 5 & 3 \end{bmatrix} \]
Solve by matrix method the system of equations:
\[ \begin{aligned} x - y + z &= 4 \\ 2x + y - 3z &= 0 \\ x + y + z &= 2 \end{aligned} \]
If
\[ A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix} \]
then verify that \( A(\text{adj} A) = |A| I \) and find \( A^{-1} \).
If \[\begin{bmatrix} 2x - y & x + 2y \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 2 & 3 \end{bmatrix}\] then the value of $x$ and $y$ will be: