What is a matrix?
A matrix is a rectangular arrangement of numbers, called elements, enclosed in brackets and organised in rows and columns. Matrices give us a compact way to store and process ordered data such as prices, scores or the coefficients of equations.
Order of a matrix
The order (or dimension) of a matrix is written as rows × columns. A matrix with $m$ rows and $n$ columns has order $m\times n$. For example, $\begin{pmatrix}2&5&1\\0&3&4\end{pmatrix}$ has 2 rows and 3 columns, so its order is $2\times 3$ and it contains $2\times 3=6$ elements. The row is read horizontally and the column vertically.
A matrix with a single row is a row matrix; one with a single column is a column matrix; one with equal numbers of rows and columns is a square matrix.
Key formula
$\text{order}=(\text{number of rows})\times(\text{number of columns}),\qquad \text{number of elements}=m\times n$
Equal matrices
Two matrices are equal only when they have the same order AND every pair of corresponding elements is equal. This condition lets us set up equations to find unknown values.
Example
If $\begin{pmatrix}x&3\\1&y\end{pmatrix}=\begin{pmatrix}4&3\\1&7\end{pmatrix}$, then by comparing corresponding positions $x=4$ and $y=7$.
The identity matrix
The identity matrix $I$ is a square matrix with $1$ on the leading diagonal and $0$ everywhere else. For order $2\times 2$, $I=\begin{pmatrix}1&0\\0&1\end{pmatrix}$. Multiplying any suitable matrix by $I$ leaves it unchanged, just as multiplying a number by $1$.
Remember
Always state the order as rows first, then columns: $2\times 3\ne 3\times 2$.