The total number of matrices $A = \left[ {\begin{array}{*{20}{c}}
0&{2x}&{2x}\\
{2y}&y&{ – y}\\
1&{ – 1}&1
\end{array}} \right];\,\left( {x,y \in R,\,x \ne y} \right)$ for which ${A^T}A = 3{I_3}$

A square matrix $A = [{a_{ij}}]$ in which ${a_{ij}} = 0$ for $i\, \ne j$ and ${a_{ij}} = k$ (constant) for $i = j$ is called a

Find the transpose of the following matrices : $\left[\begin{array}{c}5 \\ \frac{1}{2} \\ -1\end{array}\right]$.

Let $M=\left(a_{i j}\right), i, j \in\{1,2,3\}$, be the $3 \times 3$ matrix such that $a_{i j}=1$ if $j+1$ is divisible by $i$, otherwise $a_{i j}=0$. Then which of the following statements is (are) true?

($A$) $\mathrm{M}$ is invertible

($B$) There exists a nonzero column matrix $\left(\begin{array}{l}a_1 \\ a_2 \\ a_3\end{array}\right)$ such that $M\left(\begin{array}{l}a_1 \\ a_2 \\ a_3\end{array}\right)=\left(\begin{array}{l}-a_1 \\ -a_2 \\ -a_3\end{array}\right)$

($C$) The set $\left\{\mathrm{X} \in \mathbb{R}^3: \mathrm{MX}=\mathbf{0} \neq \neq 0\right\}$, where $\mathbf{0}=\left(\begin{array}{l}0 \\ 0 \\ 0\end{array}\right)$

($D$) The matrix $(M-2 I)$ is invertible, where $I$ is the $3 \times 3$ identity matrix