Let $S$ be the set containing all $3 \times 3$ matrices with entries from $\{-1, 0, 1\}$. The total number of matrices $A \in S$ such that the sum of all the diagonal elements of $A^{T}A$ is $6$ is:

If $A$ and $B$ are $3 \times 3$ matrices such that $AB = A$ and $BA = B$,then

The maximum value of $f(x) = \left|\begin{array}{ccc} \sin^{2} x & 1+\cos^{2} x & \cos 2x \\ 1+\sin^{2} x & \cos^{2} x & \cos 2x \\ \sin^{2} x & \cos^{2} x & \sin 2x \end{array}\right|, x \in R$ is:

Let $A = \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 1 \\ 3 & -1 & 0 \end{bmatrix}$ and $B$ be a matrix such that $B(I - A) = I + A$. Then the sum of the diagonal elements of $B^T B$ is equal to:

Let $\Omega$ be the set of all $3 \times 3$ symmetric matrices all of whose entries are either $0$ or $1$. Five of these entries are $1$ and four of them are $0$.
$1.$ The number of matrices in $\Omega$ is
$(A) 12$ $(B) 6$ $(C) 9$ $(D) 3$
$2.$ The number of matrices $A$ in $\Omega$ for which the system of linear equations $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ has a unique solution,is
$(A)$ less than $4$ $(B)$ at least $4$ but less than $7$ $(C)$ at least $7$ but less than $10$ $(D)$ at least $10$
$3.$ The number of matrices $A$ in $\Omega$ for which the system of linear equations $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ is inconsistent,is
$(A) 0$ $(B)$ more than $2$ $(C) 2$ $(D) 1$