Answer to Question #248987 in Linear Algebra for Almas

Question

Prove that the commutative property holds on the matrix multiplication.

Answer 1

Taking into account that

$\begin{pmatrix} 1 & -1\\ 0 & 1 \end{pmatrix} \begin{pmatrix} -1 & 1\\ 0 & 1 \end{pmatrix} = \begin{pmatrix} -1 & 0\\ 0 & 1 \end{pmatrix},$ but

$\begin{pmatrix} -1 & 1\\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & -1\\ 0 & 1 \end{pmatrix} = \begin{pmatrix} -1 & 2\\ 0 & 1 \end{pmatrix},$

we conclude that the commutative property does not hold on the matrix multiplication.