Matrix Operations

≈ 20 min

A matrix is a rectangular grid of numbers, with a size written as (rows $\times$ columns). We add matrices of the same size entry by entry, and multiply by a scalar entry by entry.

Matrix multiplication is different: to multiply $A$ ( $m\times n$ ) by $B$ ( $n\times p$ ), the inner dimensions must match. The entry in row $i$ , column $j$ of $AB$ is the dot product of row $i$ of $A$ with column $j$ of $B$ . The result is $m\times p$ .

Worked example. With $A=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ and $B=\begin{pmatrix}5\\6\end{pmatrix}$ (a $2\times 2$ times a $2\times 1$ ), the product is $2\times 1$ :

AB = \begin{pmatrix}1\cdot 5 + 2\cdot 6\\ 3\cdot 5 + 4\cdot 6\end{pmatrix} = \begin{pmatrix}17\\39\end{pmatrix}

For general matrices $A$ and $B$ , which statement is true?

$AB = BA$ always

$AB \neq BA$ in general

$AB$ is always the same size as $A$

Matrix multiplication is just entry-by-entry

If $A$ is $2\times 3$ and $B$ is $3\times 4$ , the product $AB$ has size:

$3\times 3$

$2\times 4$

$4\times 2$

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With $A=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ and $B=\begin{pmatrix}5\\6\end{pmatrix}$ , find the top entry of $AB$ .

Let $A=\begin{pmatrix}2&0\\1&3\end{pmatrix}$ and $B=\begin{pmatrix}1&4\\2&5\end{pmatrix}$ . Find the entry in row 2, column 2 of $AB$ .