Answer to Question #248991 in Linear Algebra for Almas

Question

Give an example that the commutative law does not holds on the matrix multiplication.

Answer 1

Since

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

$\begin{pmatrix} -1 & 2\\ 0 & 1 \end{pmatrix} \begin{pmatrix} 2 & -2\\ 0 & 1 \end{pmatrix} = \begin{pmatrix} -2 & 4\\ 0 & 1 \end{pmatrix},$

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