Matrix Multiplication is not Commutative/Examples

Examples illustrating Matrix Multiplication is not Commutative

Arbitrary $2 \times 2$ Matrices

Consider the matrices:

\(\ds \mathbf A\)

\(=\)

\(\ds \begin {pmatrix} 1 & 2 \\ -1 & 0 \end {pmatrix}\)

\(\ds \mathbf B\)

\(=\)

\(\ds \begin {pmatrix} 1 & -1 \\ 0 & 1 \end {pmatrix}\)

We have:

\(\ds \mathbf A \mathbf B\)

\(=\)

\(\ds \begin {pmatrix} 1 & 1 \\ -1 & 1 \end {pmatrix}\)

\(\ds \mathbf B \mathbf A\)

\(=\)

\(\ds \begin {pmatrix} 2 & 2 \\ -1 & 0 \end {pmatrix}\)

and it is seen that $\mathbf A \mathbf B \ne \mathbf B \mathbf A$.