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?

Question
Matrices
asked 2021-01-07
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?

Answers (1)

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.
0

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