If $A = \left| {\begin{array}{cccccccccccccccccccc}2&3&{ - 1}\\1&4&2\end{array}} \right|$ and $B = \left| {\begin{array}{cccccccccccccccccccc}2&3\\4&5\\2&1\end{array}} \right|$, then AB and BA are defined and equal.

Correct Answer False

Since, AB is defined.
$\therefore$ $AB = \left[ {\begin{array}{llllllllllllllllllll}2&3&{ - 1}\\1&4&2\end{array}} \right]\left[ {\begin{array}{llllllllllllllllllll}2&3\\4&5\\2&1\end{array}} \right] = \left[ {\begin{array}{llllllllllllllllllll}{14}&{20}\\{22}&{25}\end{array}} \right]$

Also BA is defined.
$\therefore BA = \left[ {\begin{array}{llllllllllllllllllll}2&3\\4&5\\2&1\end{array}} \right]\left[ {\begin{array}{llllllllllllllllllll}2&3&{ - 1}\\1&4&2\end{array}} \right]$

$= \left[ {\begin{array}{cccccccccccccccccccc}7&{18}&4\\{13}&{32}&6\\5&{10}&0\end{array}} \right]$
$\therefore$ $AB \ne BA$