Introduction to Matrices

This lesson introduces matrices: what they are, how their order is written, how to build a matrix from a rule for its entries, the main types of matrices, and how equality of matrices lets you solve for unknowns.

What a matrix is

A matrix is a rectangular arrangement of numbers (its elements or entries) set out in rows and columns. The plural of matrix is matrices. We name a matrix with a capital letter and write its entries inside square brackets, with no commas between them.

For example,

$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$

has first row $1,\ 2,\ 3$ and second row $4,\ 5,\ 6$.

Order of a matrix

The order of a matrix is its number of rows by its number of columns, written $m \times n$ and read "$m$ by $n$", where $m$ is the number of rows and $n$ the number of columns. The matrix $A$ above has $2$ rows and $3$ columns, so its order is $2 \times 3$.

A few more examples:

$B = \begin{bmatrix} 2 \\ 3 \\ 6 \end{bmatrix} \quad (\text{order } 3 \times 1), \qquad C = \begin{bmatrix} 2 & -3 & 5 \end{bmatrix} \quad (\text{order } 1 \times 3)$

If the order is $m \times n$, the total number of entries is $mn$. So a $2 \times 3$ matrix has $2 \times 3 = 6$ entries.

General entry notation

A general matrix is written $A = [a_{ij}]$, where $a_{ij}$ is the entry in row $i$ and column $j$, with $1 \le i \le m$ and $1 \le j \le n$ and $i, j \in \mathbb{N}$.

Order from a given number of entries. If a matrix has $6$ entries, its order can be any pair of factors of $6$: $1 \times 6$, $2 \times 3$, $3 \times 2$ or $6 \times 1$.

Building a matrix from a formula

When a rule for $a_{ij}$ is given, evaluate it for each position.

Example: $a_{ij} = i + j$, order $3 \times 2$

$a_{11} = 1+1 = 2, \quad a_{12} = 1+2 = 3$ $a_{21} = 2+1 = 3, \quad a_{22} = 2+2 = 4$ $a_{31} = 3+1 = 4, \quad a_{32} = 3+2 = 5$

so

$A = \begin{bmatrix} 2 & 3 \\ 3 & 4 \\ 4 & 5 \end{bmatrix}$

Example: $a_{ij} = \dfrac{(i+j)^2}{2}$, order $2 \times 2$

$a_{11} = \frac{(1+1)^2}{2} = \frac{4}{2} = 2, \quad a_{12} = \frac{(1+2)^2}{2} = \frac{9}{2}$ $a_{21} = \frac{(2+1)^2}{2} = \frac{9}{2}, \quad a_{22} = \frac{(2+2)^2}{2} = \frac{16}{2} = 8$

so

$A = \begin{bmatrix} 2 & \tfrac{9}{2} \\ \tfrac{9}{2} & 8 \end{bmatrix}$

Types of matrices

• Row matrix: a single row, e.g. $\begin{bmatrix} 0 & 1 & 2 \end{bmatrix}$ (order $1 \times 3$).
• Column matrix: a single column, e.g. $\begin{bmatrix} 3 \\ 4 \end{bmatrix}$ (order $2 \times 1$).
• Square matrix: equal numbers of rows and columns, e.g. $\begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix}$.
• Diagonal matrix: a square matrix in which every entry off the main diagonal is $0$, e.g. $\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$ or $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{bmatrix}$.
• Scalar matrix: a diagonal matrix whose diagonal entries are all equal, e.g. $\begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}$.
• Identity matrix: a diagonal matrix whose diagonal entries are all $1$. We write $I_1, I_2, I_3$ for the identity matrices of order $1, 2, 3$, for example $I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ and $I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$.
• Zero matrix: every entry is $0$.

Equality of matrices

Two matrices are equal when they have the same order and their corresponding entries are equal. This lets you read off unknowns by matching position to position.

Example: matching entries

$\begin{bmatrix} 2 & 4 \\ 3 & 6 \end{bmatrix} = \begin{bmatrix} x & y \\ 3 & z \end{bmatrix}$

gives $x = 2$, $y = 4$, $z = 6$.

Example: leading to a quadratic

$\begin{bmatrix} x+y & 5+z \\ xy & 8 \end{bmatrix} = \begin{bmatrix} 6 & 3 \\ 8 & 8 \end{bmatrix}$

Matching entries gives $5 + z = 3$, so $z = -2$, together with $x + y = 6$ and $xy = 8$. From $x = 6 - y$,

$(6 - y)\,y = 8 \implies y^2 - 6y + 8 = 0 \implies (y-2)(y-4) = 0$

so $y = 2$ or $y = 4$, giving $x = 4$ or $x = 2$.

Key takeaways

• A matrix is a rectangular array of numbers, and its order is rows by columns; an $m \times n$ matrix has $mn$ entries.
• Given a rule $a_{ij}$, evaluate it at each position to build the matrix.
• The main types are row, column, square, diagonal, scalar, identity and zero matrices.
• Two matrices are equal exactly when they have the same order and equal corresponding entries, which you can use to solve for unknowns.