Let $G(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$. If $x+y=0$,then $G(x) G(y) =$

If $A = \begin{bmatrix} a & b \\ c & -a \end{bmatrix}$ is such that $A^2 = I$, then . . . . . . .

Construct a $3 \times 2$ matrix whose elements are given by $a_{ij} = \frac{1}{2}|i - 3j|$.

Let $A = \begin{bmatrix} x & 1 \\ 1 & 0 \end{bmatrix}$,$x \in \mathbb{R}$ and $A^{4} = [a_{ij}]$. If $a_{11} = 109$,then $a_{22}$ is equal to

If $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} x & y \\ 0 & x \end{bmatrix}$,then $AB = BA$ (given $B \neq I$). Which of the following matrices $B$ satisfies this condition?

Show that $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{bmatrix} \begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4 \end{bmatrix} \ne \begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{bmatrix}$