York

2021-01-07

Let $A=\left[\begin{array}{cc}-5& -2\\ 1& 2\end{array}\right],B=\left[\begin{array}{cc}-3& -5\\ 1& 5\end{array}\right]$
If possible , compute the following . If an answer does not exist , enter DNE.
AB,BA-?
True or False: AB=BA?

lamusesamuset

Step 1
$A=\left[\begin{array}{cc}-5& -2\\ 1& 2\end{array}\right],B=\left[\begin{array}{cc}-3& -5\\ 1& 5\end{array}\right]$
$AB=\left[\begin{array}{cc}-5& -2\\ 1& 2\end{array}\right]\left[\begin{array}{cc}-3& -5\\ 1& 5\end{array}\right]$
$=\left[\begin{array}{cc}15-2& 25-10\\ -3+2& -5+10\end{array}\right]$
$=\left[\begin{array}{cc}13& 15\\ -1& 5\end{array}\right]$
Step 2
$A=\left[\begin{array}{cc}-5& -2\\ 1& 2\end{array}\right],B=\left[\begin{array}{cc}-3& -5\\ 1& 5\end{array}\right]$
$BA=\left[\begin{array}{cc}-3& -5\\ 1& 5\end{array}\right]×\left[\begin{array}{cc}-5& -2\\ 1& 2\end{array}\right]$
$=\left[\begin{array}{cc}15-5& 6-10\\ -5+5& -10+10\end{array}\right]$
$=\left[\begin{array}{cc}10& -4\\ 0& 0\end{array}\right]$
Step 3
For every pair of square matrices A and B of the same size, $AB\ne BA$,because matrix multiplication is not commutative.
Given statement in the question is False.

Jeffrey Jordon