Question

If $A=\left[\begin{array}{ccc}1& -2& 3\\ -4& 2& 5\end{array}\right]$ and $B=\left[\begin{array}{cc}2& 3\\ 4& 5\\ 2& 1\end{array}\right]$, then the product of AB and BA is

• A. $\left[\begin{array}{ccc}-10& 2& 21\\ -16& 2& 37\\ -2& -2& 11\end{array}\right]$
• B. $\left[\begin{array}{cc}0& -4\\ 10& 3\end{array}\right]$
• C. $\left[\begin{array}{cc}-64& -8\\ -148& 26\end{array}\right]$
• D. $Cannotbecomputed$

Answer 1

Answer: (D)

$Cannotbecomputed$

$AB=\left[\begin{array}{ccc}1& -2& 3\\ -4& 2& 5\end{array}\right]\left[\begin{array}{cc}2& 3\\ 4& 5\\ 2& 1\end{array}\right]$
$\left[\begin{array}{cc}2-8+6& 3-10+3\\ -8+8+10& -12+10+5\end{array}\right]=\left[\begin{array}{cc}0& -4\\ 10& 3\end{array}\right]$
$BA=\left[\begin{array}{cc}2& 3\\ 4& 5\\ 2& 1\end{array}\right]\left[\begin{array}{ccc}1& -2& 3\\ -4& 2& 5\end{array}\right]$
$\left[\begin{array}{ccc}2-12& -4+6& 6+15\\ 4-20& -8+10& 12+25\\ 2-4& -4+2& 6+5\end{array}\right]=\left[\begin{array}{ccc}-10& 2& 21\\ -16& 2& 37\\ -2& -2& 11\end{array}\right]$

Thus, order of AB is $2\times 2$ and order of $BA$ is $3\times3.$

So, the product AB and BA can not be computed.