Exercise 508Linear algebra › Matrices and linear systemsHigh schoolAdded 3 Nov 2026

Matrix multiplication

#matrix multiplication#non-commutativity

Statement

Let
$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \qquad B = \begin{pmatrix} 2 & 0 \\ 1 & -1 \end{pmatrix}.$
Compute $AB$ and $BA$ . Are they equal?

Hint

The entry $(i,j)$ of $AB$ is the dot product of row $i$ of $A$ with column $j$ of $B$ .

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Solution

Product $AB$ .
$AB = \begin{pmatrix} 1\cdot2+2\cdot1 & 1\cdot0+2\cdot(-1) \\ 3\cdot2+4\cdot1 & 3\cdot0+4\cdot(-1) \end{pmatrix} = \begin{pmatrix} 4 & -2 \\ 10 & -4 \end{pmatrix}.$

Product $BA$ .
$BA = \begin{pmatrix} 2\cdot1+0\cdot3 & 2\cdot2+0\cdot4 \\ 1\cdot1+(-1)\cdot3 & 1\cdot2+(-1)\cdot4 \end{pmatrix} = \begin{pmatrix} 2 & 4 \\ -2 & -2 \end{pmatrix}.$

Conclusion. $AB \neq BA$ : matrix multiplication is not commutative.

Matrix multiplication

Statement

Solution

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