Let $A$ be a symmetric matrix of order $2$ with integer entries. If the sum of the diagonal elements of $A ^{2}$ is $1,$ then the possible number of such matrices is

Show that the matrix $B ^{\prime}A B$ is symmetric or skew symmetric according as $A$ is symmetric or skew symmetric.

If $A=\left[\begin{array}{cc}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{array}\right], B=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right], C=\mathrm{ABA}^{\mathrm{T}}$ and $\mathrm{X}$ $=\mathrm{A}^{\mathrm{T}} \mathrm{C}^2 \mathrm{~A}$, then $\operatorname{det} \mathrm{X}$ is equal to :

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 ${A^T}A = 3{I_3}$

Let $X$ and $Y$ be two arbitrary, $3 \times 3$, non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$, nonzero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

$(A)$ $Y^3 Z^4-Z^4 Y^3$ $(B)$ $X ^{44}+ Y ^{44}$

$(C)$ $X ^4 Z ^3- Z ^3 X ^4$ $(B)$ $X ^{23}+ Y ^{23}$