Matrix Multiplication is not Commutative/Examples
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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 $\mathbf A \mathbf B \ne \mathbf B \mathbf A$.