Question

Let $A=\left[\begin{array}{cc}3& 7\\ 2& 5\end{array}\right]$ & $B=\left[\begin{array}{cc}6& 8\\ 7& 9\end{array}\right]$ check whether $AB=BA$ or not.

Answer 1

Given,

$A=\left[\begin{array}{cc}3& 7\\ 2& 5\end{array}\right]$ & $B=\left[\begin{array}{cc}6& 8\\ 7& 9\end{array}\right]$.

Now,

$AB=$ $\left[\begin{array}{cc}3& 7\\ 2& 5\end{array}\right]$ $\left[\begin{array}{cc}6& 8\\ 7& 9\end{array}\right]$

or, $AB=$ $\left[\begin{array}{cc}18+49& 24+63\\ 12+35& 16+45\end{array}\right]$

or, $AB=$ $\left[\begin{array}{cc}67& 87\\ 47& 61\end{array}\right]$......(1).

And,

$BA=$ $\left[\begin{array}{cc}6& 8\\ 7& 9\end{array}\right]$ $\left[\begin{array}{cc}3& 7\\ 2& 5\end{array}\right]$

or, $BA=$ $\left[\begin{array}{cc}18+16& 42+40\\ 21+18& 49+45\end{array}\right]$

or, $BA=$ $\left[\begin{array}{cc}34& 82\\ 39& 94\end{array}\right]$.......(2).

Clearly from (1) and (2) we get,

$AB\ne BA$.