Question

Let $A$ be a $2 \times 3$ matrix whereas $B$ be a $3 \times 2$ matrix. If det. $(A B)=4,$ then the value of det. $(B A)$, is ____

Answer 1

$Let A=\begin{bmatrix}a_{1}&a_{2}&a_{3}\\ a_{4}&a_{5}&a_{5}\end{bmatrix}B=\begin{bmatrix}b_{1}&b_{2}\\ b_{3}&b_{4}\\ b_{5}&b_{6}\end{bmatrix} $
So, det $(BA)$
$=\begin{vmatrix}b_{1}a_{1}+b_{2}a_{4}&b_{1}a_{2}+b_{2}a_{5}&b_{1}a_{3}+b_{2}a_{5}\\ b_{3}a_{1}+b_{4}a_{4}&b_{3}a_{2}+b_{4}a_{5}&b_{3}a_{3}+b_{4}a_{5}\\ b_{5}a_{1}+b_{6}a_{4}&b_{5}a_{2}+b_{6}a_{5}&b_{5}a_{3}+b_{6}a_{5}\end{vmatrix} $
$=\begin{vmatrix}a_{1}&a_{2}&a_{3}\\ a_{4}&a_{5}&a_{6}\\ 0&0&0\end{vmatrix}\times\begin{vmatrix}b_{1}&b_{2}&0\\ b_{3}&b_{4}&0\\ b_{5}&b_{6}&0\end{vmatrix}=0$
(column by row)