The number of matrices with order $3 \times 2$ whose each entry is $1$ or $2$ is . . . . . . .

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$ is a matrix satisfying the equation $A A^T = 9 I$,where $I$ is the identity matrix,then $a^2 + b^2 =$

Out of the following,a skew-symmetric matrix is:

Compute the following: $\left[ {\begin{array}{cc} {{\cos }^2}x & {{\sin }^2}x \\ {{\sin }^2}x & {{\cos }^2}x \end{array}} \right] + \left[ {\begin{array}{cc} {{\sin }^2}x & {{\cos }^2}x \\ {{\cos }^2}x & {{\sin }^2}x \end{array}} \right]$

If $A^{\prime}=\begin{bmatrix}-2 & 3 \\ 1 & 2\end{bmatrix}$ and $B=\begin{bmatrix}-1 & 0 \\ 1 & 2\end{bmatrix},$ then find $(A+2B)^{\prime}$.

Consider the following information regarding the number of men and women workers in three factories $I, II$ and $III$.

Factory	Men and Women Workers
$I$	$30$ Men,$25$ Women
$II$	$25$ Men,$31$ Women
$III$	$27$ Men,$26$ Women

Represent the above information in the form of a $3 \times 2$ matrix. What does the entry in the third row and second column represent?