# Matrix Multiplication is not Commutative/Examples/Arbitrary 2x2 Matrices

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## Example illustrating Matrix Multiplication is not Commutative

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 $\mathbf A \mathbf B \ne \mathbf B \mathbf A$.