Question

$2\times 2$ matrix where elements come from $\{p,q,r,s\}$ without repeating satisfying following conditions :-
$a_{11}\ne p$
$a_{12}\ne q$
$a_{21}\ne r$
$a_{22}\ne s$

Answer 1

The elements of $2\times 2$ matrix are $\{p,q,r,s\}$

But

$a_{11}\ne p$

$a_{12}\ne q$

$a_{21}\ne r$

$a_{22}\ne s$

So there are $9$ possibilities

The matrices can be from one the the following.

$\left[\begin{array}{c}qp\\ sr\end{array}\right]$

$\left[\begin{array}{c}qr\\ sp\end{array}\right]$

$\left[\begin{array}{c}qs\\ pr\end{array}\right]$

$\left[\begin{array}{c}rp\\ sq\end{array}\right]$

$\left[\begin{array}{c}rs\\ pq\end{array}\right]$

$\left[\begin{array}{c}rs\\ qp\end{array}\right]$

$\left[\begin{array}{c}sp\\ qr\end{array}\right]$

$\left[\begin{array}{c}sr\\ pq\end{array}\right]$

$\left[\begin{array}{c}sr\\ qp\end{array}\right]$