Question

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

Answer 1

Answer: 0

$LetA=\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ a_4& a_5& a_5\end{array}\right]B=\left[\begin{array}{cc}{b}_1& b_2\\ b_3& b_4\\ b_5& b_6\end{array}\right]$
So, det $(BA)$
$=\left|\begin{array}{ccc}{b}_1{a}_1+b_2{a}_4& b_1{a}_2+b_2{a}_5& b_1{a}_3+b_2{a}_5\\ b_3{a}_1+b_4{a}_4& b_3{a}_2+b_4{a}_5& b_3{a}_3+b_4{a}_5\\ b_5{a}_1+b_6{a}_4& b_5{a}_2+b_6{a}_5& b_5{a}_3+b_6{a}_5\end{array}\right|$
$=\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ a_4& a_5& a_6\\ 0& 0& 0\end{array}\right|\times \left|\begin{array}{ccc}{b}_1& b_2& 0\\ b_3& b_4& 0\\ b_5& b_6& 0\end{array}\right|=0$
(column by row)