If $P$ is a $3 \times 3$ real matrix such that $P ^{ T }=a P +( a -1) I$, where $a > 1$, then $……….$

If $A = \left( {\begin{array}{*{20}{c}}2&{ – 1}\\{ – 1}&2\end{array}} \right)$ and $I$ is the unit matrix of order $2$, then ${A^2}$ equals

If $A,B,C$ are three square matrices such that $AB = AC$ implies $B = C$, then the matrix $A$  is always a/an

If $A = \left[ {\begin{array}{*{20}{c}}0&1\\0&0\end{array}} \right],I$ is the unit matrix of order $2$ and $a, b$ are arbitrary constants, then ${(aI + bA)^2}$ is equal to

Show that

$\left[ {\begin{array}{*{20}{c}}
  5&{ – 1} \\ 
  6&7 
\end{array}} \right]\left[ {\begin{array}{*{20}{l}}
  2&1 \\ 
  3&4 
\end{array}} \right]$ $ \ne \left[ {\begin{array}{*{20}{l}}
  2&1 \\ 
  3&4 
\end{array}} \right]\left[ {\begin{array}{*{20}{l}}
  5&{ – 1} \\ 
  6&7 
\end{array}} \right]$