Question

# Let A=begin{bmatrix}-5 & -2 1 & 2 end{bmatrix} ,B=begin{bmatrix}-3 & -5 1 & 5 end{bmatrix} If possible , compute the following . If an answer does not exist , enter DNE. AB,BA-? True or False: AB=BA?

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

2021-01-08
Step 1
$$A=\begin{bmatrix}-5 & -2 \\1 & 2 \end{bmatrix} ,B=\begin{bmatrix}-3 & -5 \\1 & 5 \end{bmatrix}$$
$$AB=\begin{bmatrix}-5 & -2 \\1 & 2 \end{bmatrix}\begin{bmatrix}-3 & -5 \\1 & 5 \end{bmatrix}$$
$$=\begin{bmatrix}15-2 & 25-10 \\-3+2 & -5+10 \end{bmatrix}$$
$$=\begin{bmatrix}13 & 15 \\-1 & 5 \end{bmatrix}$$
Step 2
$$A=\begin{bmatrix}-5 & -2 \\1 & 2 \end{bmatrix} ,B=\begin{bmatrix}-3 & -5 \\1 & 5 \end{bmatrix}$$
$$BA=\begin{bmatrix}-3 & -5 \\1 & 5 \end{bmatrix}\times\begin{bmatrix}-5 & -2 \\1 & 2 \end{bmatrix}$$
$$=\begin{bmatrix}15-5 & 6-10 \\-5+5 & -10+10 \end{bmatrix}$$
$$=\begin{bmatrix}10 & -4 \\0 & 0 \end{bmatrix}$$
Step 3
For every pair of square matrices A and B of the same size, $$AB \neq BA$$,because matrix multiplication is not commutative.
Given statement in the question is False.