Matrix Product¶

Definition¶

If $\boldsymbol{A}$ has size $m\times n$ and $\boldsymbol{B}$ has size $n\times k$$k$, the matrix product $\boldsymbol{A}\boldsymbol{B}$ is the $m\times k$ matrix with entries $\displaystyle (\boldsymbol{A}\boldsymbol{B})_{ij}=\sum_{r=1}^n a_{ir}b_{rj}$$\sum _{r=1}^n{a}_{ir}{b}_{rj}$.

Interpretation¶

Matrix multiplication combines rows of the left matrix with columns of the right matrix and represents composition of linear maps: $(\boldsymbol{A}\boldsymbol{B})\boldsymbol{x}=\boldsymbol{A}(\boldsymbol{B}\boldsymbol{x})$$\mathbf{A}(\mathbf{B}\mathbf{x})$. The row-times-column rule is exactly what makes this hold.

The entry $c_{22}=a_{21}b_{12}+a_{22}b_{22}+a_{23}b_{32}$ is obtained by multiplying row $2$ of $\boldsymbol{A}$ by column $2$ of $\boldsymbol{B}$. Both have three entries, which is why the inner sizes of the $2\times3$ and $3\times2$ matrices must agree; the outer sizes give the $2\times2$ size of the product.

A second reading uses columns. If $\boldsymbol{a}_1,\ldots,\boldsymbol{a}_n$ are the columns of $\boldsymbol{A}$, then $\boldsymbol{A}\boldsymbol{x}=x_1\boldsymbol{a}_1+\cdots+x_n\boldsymbol{a}_n$ is a linear combination of these columns, and column $j$ of $\boldsymbol{A}\boldsymbol{B}$ is $\boldsymbol{A}$ times column $j$ of $\boldsymbol{B}$. In the second Example, $3\begin{pmatrix}1\\0\end{pmatrix}+4\begin{pmatrix}2\\1\end{pmatrix}=\begin{pmatrix}11\\4\end{pmatrix}$$\left(\begin{array}{c}11\\ 4\end{array}\right)$. With this reading, a linear system $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ asks for a combination of the columns of $\boldsymbol{A}$ that equals $\boldsymbol{b}$.

The order of the factors matters. When $\boldsymbol{A}\boldsymbol{B}$ is defined, $\boldsymbol{B}\boldsymbol{A}$ may not be, and even for square matrices the two usually differ: for $\boldsymbol{A}=\begin{pmatrix}0&1\\0&0\end{pmatrix}$ and $\boldsymbol{B}=\begin{pmatrix}0&0\\1&0\end{pmatrix}$$\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)$, $\boldsymbol{A}\boldsymbol{B}=\begin{pmatrix}1&0\\0&0\end{pmatrix}$ but $\boldsymbol{B}\boldsymbol{A}=\begin{pmatrix}0&0\\0&1\end{pmatrix}$$\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)$. A common slip is to multiply matrices entry by entry, as for matrix addition; that is a different operation.

Key Results¶

• The product $\boldsymbol{A}\boldsymbol{B}$ is defined only when the number of columns of $\boldsymbol{A}$ equals the number of rows of $\boldsymbol{B}$.
• Matrix multiplication is associative and distributive, but not generally commutative.

Examples¶

• A $2\times3$ matrix times a $3\times4$ matrix gives a $2\times4$ matrix.
• $\begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}3\\4\end{pmatrix}=\begin{pmatrix}11\\4\end{pmatrix}$$\left(\begin{array}{c}11\\ 4\end{array}\right)$.

Prerequisites¶

• Matrix

Related Concepts¶

• Product of Linear Maps
• Linear Map
• Identity Matrix
• Vector
• Distributivity

Sources¶

• Math II: Linear Algebra · Week 3
• Math V: Advanced Linear Algebra · Week 1 (revisited)

Aliases: matrix multiplication, product of matrices Symbols: sum

Last updated: 20 October 2026 03:02