Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix}$ and $B = A^{20}$. Then the sum of the elements of the first column of $B$ is?

For $A = \begin{bmatrix} 0 & 0 & 3 \\ 0 & 3 & 0 \\ 3 & 0 & 0 \end{bmatrix}$,which statement is correct?

The number of all $3 \times 3$ matrices $A$,with entries from the set $\{-1, 0, 1\}$ such that the sum of the diagonal elements of $AA^{T}$ is $3$,is

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?

If $F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$,show that $F(x) F(y) = F(x+y)$.

Evaluate $A^2+2I$ if $A=\begin{bmatrix} 1 & 0 \\ 1 & 2 \end{bmatrix}$. (in $A$)