If for the matrix, $A =\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right], AA ^{ T }= I _{2},$ then the value of $\alpha^{4}+\beta^{4}$ is ……. .

If matrix $A = {\left[ {{a_{ij}}} \right]_{3 \times 3}} , B = {\left[ {{b_{ij}}} \right]_{3 \times 3}}$ , where $a_{ij} + a_{ji} = 0$ and $b_{ij} -b_{ji} = 0\, \forall\, i , j$ then $A^4B^3$ is

For the matrices $\mathrm{A}$ and $\mathrm{B}$, verify that $(\mathrm{A} \mathrm{B})^{\prime}=\mathrm{B}^{\prime} \mathrm{A}^{\prime}$  where  $\mathrm{A}=\left[\begin{array}{l}0 \\ 1 \\ 2\end{array}\right], \mathrm{B}=\left[\begin{array}{lll}1 & 5 & 7\end{array}\right]$.

Let $A$ be a $2 \times 2$ real matrix with entries from $\{0,1\}$ and $|\mathrm{A}| \neq 0 .$ Consider the following two statements :

$(P)$ If $A \neq I_{2},$ then $|A|=-1$

$(\mathrm{Q})$ If $|\mathrm{A}|=1,$ then $\operatorname{tr}(\mathrm{A})=2$

where $I_{2}$ denotes $2 \times 2$ identity matrix and $\operatorname{tr}(A)$ denotes the sum of the diagonal entries of $A$ Then

Let $A$ and $B$ be any two $3 \times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is $NOT$ true?