Let matrix $A = \left[ {\begin{array}{*{20}{c}}
  1&2&3 \\ 
  0&5&4 \\ 
  0&3&2 
\end{array}} \right]$ and $A^3 -8A^2 + \alpha A + \beta I = O$ then ordered pair $(\alpha , \beta)$ is

Let $A$ and $B$ be two $3 \times 3$ real matrices such that $\left(A^{2}-B^{2}\right)$ is invertible matrix. If $A^{5}=B^{5}$ and $A^{3} B^{2}=A^{2} B^{3}$, then the value of the determinant of the matrix $A^{3}+B^{3}$ is equal to:

Let $A =\left(\begin{array}{ll}2 & -2 \\ 1 & -1\end{array}\right)$ and $B =\left(\begin{array}{ll}-1 & 2 \\ -1 & 2\end{array}\right)$. Then the number of elements in the set $\left\{( n , m ): n , m \in\{1,2, \ldots . .10\}\right.$ and $\left.nA ^{ n }+ mB ^{ m }= I \right\}$ is

If $A = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right],B = \left[ {\begin{array}{*{20}{c}}0&{ – i}\\i&0\end{array}} \right]$ then ${(A + B)^2}$ equals

If $A,B$are square matrices of order $3, A$ is non- singular and $AB = O$, then $B$ is a