For two $3\times3$ matrices $A$ and $B$ , let $A+ B\, = 2B'$ and $3A + 2B\, = I_3$, where $B'$ is the transpose of $B$ and $I_3$ is $3\times3$ identity matrix. Then

Find the values of $x, y, z$ if the matrix $A=\left[\begin{array}{ccc}0 & 2 y & z \\ x & y & -z \\ x & -y & z\end{array}\right]$ satisfy the equation $A^{\prime} A=I$.

Let $A =\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$ and $B =\left[\begin{array}{l}\alpha \\ \beta\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0\end{array}\right]$ such that
$AB = B$ and $a + d =2021,$ then the value of $ad – bc$ is equal to …… .

Express the following matrices as the sum of a symmetric and a skew symmetric matrix : $\left[\begin{array}{cc}1 & 5 \\ -1 & 2\end{array}\right]$

Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by $tr(A),$ the sum of diagonal entries of $a$. Assume that ${A^2} = I$ .

Statement $-1 :$ If $A \ne I,A \ne – I$ then $\det \left( A \right) = – 1$

Statement $-2 :$ If $A \ne I,A \ne – I$ then ${\rm{tr}}\left( A \right) \ne 0$