Skill: Multiply two matrices

To multiply matrices, the number of columns in the first matrix must equal the number of rows in the second. Each entry in the product comes from multiplying one row by one column, then adding the results.

For example, let

$$A=\begin{bmatrix} 1&2&0\\ -1&3&4 \end{bmatrix}, \qquad B=\begin{bmatrix} 2&1\\ 0&-2\\ 5&3 \end{bmatrix}.$$

Matrix $A$ is $2\times3$ and $B$ is $3\times2$. The inner dimensions match, so $AB$ is defined; its dimensions are the outer dimensions, $2\times2$.

Calculate each entry using a row of $A$ and a column of $B$:

$$AB= \begin{bmatrix} 1(2)+2(0)+0(5) & 1(1)+2(-2)+0(3)\\ (-1)(2)+3(0)+4(5) & (-1)(1)+3(-2)+4(3) \end{bmatrix} = \begin{bmatrix} 2&-3\\ 18&5 \end{bmatrix}.$$

The order matters: reversing the factors gives a $3\times3$ product, not the same $2\times2$ matrix.