Question

Find $AB$, if $A=\left[\begin{array}{ccc}1& 2& 3\\ 1& -2& 3\end{array}\right]$ and $B=\left[\begin{array}{cc}1& -1\\ 1& 2\\ 1& -2\end{array}\right]$
Examine whether $AB$ has inverse or not.

Answer 1

$A=\left[\begin{array}{ccc}1& 2& 3\\ 1& -2& 3\end{array}\right]$ $B=\left[\begin{array}{cc}1& -1\\ 1& 2\\ 1& -2\end{array}\right]$

$\Rightarrow AB=\left[\begin{array}{ccc}1& 2& 3\\ 1& -2& 3\end{array}\right]\left[\begin{array}{cc}1& -1\\ 1& 2\\ 1& -2\end{array}\right]$

$=\left[\begin{array}{cc}(1\times 1)+(2\times 1)+(3\times 1)& (1\times -1)+(2\times 2)+(3\times -2)\\ (1\times 1)+(-2\times 1)+(3\times 1)& (1\times -1)+(-2\times 2)+(3\times -2)\end{array}\right]$

$=\left[\begin{array}{cc}6& -3\\ 2& -11\end{array}\right]$

$\Rightarrow \left|AB\right|=-66+6=-60$

$\therefore \left|AB\right|\ne 0$

$\therefore AB$ has inverse.