Question

If A = $\left[\begin{array}{ccc}1& 2& 5\\ 3& 4& 6\end{array}\right]$, B = $\left[\begin{array}{cc}4& 0\\ 2& 1\\ 1& 5\end{array}\right]$ then prove that $AB\ne BA$.

Answer 1

Given $A=\left[\begin{array}{ccc}1& 2& 5\\ 3& 4& 6\end{array}\right]$ and $B=\left[\begin{array}{cc}4& 0\\ 2& 1\\ 1& 5\end{array}\right]$

To prove: $AB\ne BA$

$AB=\left[\begin{array}{ccc}1& 2& 5\\ 3& 4& 6\end{array}\right]\left[\begin{array}{cc}4& 0\\ 2& 1\\ 1& 5\end{array}\right]=\left[\begin{array}{cc}4+4+5& 0+2+25\\ 12+8+6& 0+4+30\end{array}\right]=\left[\begin{array}{cc}13& 27\\ 26& 34\end{array}\right]...\left(1\right)$

$BA=\left[\begin{array}{cc}4& 0\\ 2& 1\\ 1& 5\end{array}\right]\left[\begin{array}{ccc}1& 2& 5\\ 3& 4& 6\end{array}\right]=\left[\begin{array}{ccc}4+0& 8+0& 20+0\\ 2+3& 4+4& 10+6\\ 1+15& 2+20& 5+30\end{array}\right]=\left[\begin{array}{ccc}4& 8& 20\\ 5& 8& 16\\ 16& 22& 35\end{array}\right]...\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$ we can say that $AB\ne BA$