Linear and multilinear algebra; matrix theory Counterexample

Non-commutativity of matrix multiplication

Counterexample

Let $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ , as real $2 \times 2$ matrices. Then $AB = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \qquad BA = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.$ Thus $AB \ne BA$ . This shows that matrix multiplication of $n \times n$ matrices is not commutative for $n = 2$ .

Remarks

The same example, extended with zero rows and zero columns, shows that $M_n(R)$ is not commutative for each $n \ge 2$ and each nonzero ring $R$ . Two matrices $A$ and $B$ with $AB = BA$ are said to commute. Some pairs commute, for example a matrix and the identity matrix, or two diagonal matrices.

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Matrix definition
Matrix multiplication definition
Non-commutativity of matrix multiplication counterexample

Tag	`0P506SPZP2`
URI	`m2://0P506SPZP2`
Kind	counterexample
Writing	W0 stub
MSC	15A27
Uses	Matrix multiplication

Non-commutativity of matrix multiplication

Counterexample

Remarks

Reading path

Further reading

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