Chapter 2: The algebra of matrices

Definition

An $n\times m$ matrix is a grid of numbers with $n$ rows and $m$ columns: \[ A=\begin{bmatrix}a_{11}&a_{12}&\dots&a_{1m}\\a_{21}&a_{22}&\dots&a_{2m}\\\vdots&\vdots&&\vdots\\a_{n1}&a_{n2}&\dots&a_{nm}\end{bmatrix}\]

The $(i,j)$ entry of a matrix $A$ is $a_{ij}$, the number in row $i$ and column $j$ of $A$.

Examples

If $B=\begin{bmatrix} 99&3&5\\7&-20&14\end{bmatrix}$, then $B$ is a $2\times 3$ matrix, and the $(1,1)$ entry of $B$ is $b_{11}=99$, the $(1,3)$ entry of $B$ is $b_{13}=5$, the $(2,1)$ entry is $b_{21}=7$, etc.
$\begin{bmatrix}3\\2\\4\\0\\-1\end{bmatrix}$ is a $5\times 1$ matrix. A matrix like this with one column is called a column vector.
$\begin{bmatrix}3&2&4&0&-1\end{bmatrix}$ is a $1\times 5$ matrix. A matrix like this with one row is called a row vector.

Even though the row matrix and the column matrix above have the same entries, they have a different “shape”, or “size”, so we must think of them has being different matrices. Let's give the definitions to make this precise.

Definition

Two matrices $A$ and $B$ have the same size if they have the same number of rows, and they have the same number of columns.

If two matrices do not have the same size, we say they have different sizes.

Definition

Two matrices $A$ and $B$ are said to be equal if both of the following conditions hold:

$A$ and $B$ have the same size; and
every entry of $A$ is equal to the corresponding entry of $B$; in other words, for every $(i,j)$ so that $A$ and $B$ have an $(i,j)$ entry, we have $a_{ij}=b_{ij}$.

When $A$ and $B$ are equal matrices, we write $A=B$. Otherwise, we write $A\ne B$.

Examples

$\begin{bmatrix}3\\2\\4\\0\\-1\end{bmatrix}\ne \begin{bmatrix}3&2&4&0&-1\end{bmatrix}$, since these matrices have different sizes: the first is $5\times 1$ but the second is $1\times 5$.

$\begin{bmatrix}1\\2\end{bmatrix}\ne\begin{bmatrix}1 &0\\2&0\end{bmatrix}$ since these matrices are not the same size.
$\begin{bmatrix}1&0\\0&1\end{bmatrix}\ne \begin{bmatrix}1&0\\1&0\end{bmatrix}$ because even though they have the same size, the $(2,1)$ entries are different.
If $\begin{bmatrix}3x&7y+2\\8z-3&w^2\end{bmatrix}=\begin{bmatrix}1&2z\\\sqrt2&9\end{bmatrix}$ then we know that all the corresponding entries are equal, so we get four equations:\begin{align*}3x&=1\\7y+2&=2z\\8z-3&=\sqrt2\\w^2&=9\end{align*}