Further Maths/Lessons/Matrices/Introduction to Matrices
Core Pure · Matrices · Year 1

Introduction to Matrices

A rectangular grid of numbers with its own arithmetic — addition, subtraction, and scalar multiplication all behave exactly as you'd hope.

14 min read 3 worked examples

What is a matrix?

Definition 18.1 — matrix and order

A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with $m$ rows and $n$ columns has order $m\times n$. The entry in row $i$, column $j$ is written $a_{ij}$. Two matrices are equal only if they have the same order and every corresponding entry is equal.

Addition, subtraction, and scalar multiplication

Method 18.2

Two matrices can be added or subtracted only if they have the same order — add or subtract corresponding entries. Multiplying a matrix by a scalar $k$ multiplies every entry by $k$.

$$\begin{pmatrix}1&2\\3&4\end{pmatrix}+\begin{pmatrix}0&1\\-1&2\end{pmatrix}=\begin{pmatrix}1&3\\2&6\end{pmatrix},$$ $$2\begin{pmatrix}1&2\\3&4\end{pmatrix}=\begin{pmatrix}2&4\\6&8\end{pmatrix}$$
Example 1 · Combining matrix operations
Given $A=\begin{pmatrix}3&-1\\2&5\end{pmatrix}$ and $B=\begin{pmatrix}0&4\\-2&1\end{pmatrix}$, find $2A-3B$.

$2A=\begin{pmatrix}6&-2\\4&10\end{pmatrix}$, $3B=\begin{pmatrix}0&12\\-6&3\end{pmatrix}$.

$$2A-3B=\begin{pmatrix}6-0&-2-12\\4-(-6)&10-3\end{pmatrix}=\begin{pmatrix}6&-14\\10&7\end{pmatrix}$$
Example 2 · Solving a matrix equation for unknowns
Given $\begin{pmatrix}x&3\\1&y\end{pmatrix}+\begin{pmatrix}2&-1\\4&5\end{pmatrix}=\begin{pmatrix}7&2\\5&8\end{pmatrix}$, find $x$ and $y$.

Comparing entries: $x+2=7 \Rightarrow x=5$; $y+5=8 \Rightarrow y=3$.

Special matrices: square, zero, and identity

Definition 18.3 — square, zero, and identity matrices

A matrix is square if its number of rows equals its number of columns. The zero matrix $0$ has every entry equal to $0$; adding it to any matrix of the same order leaves that matrix unchanged. The identity matrix $I_n$ is the $n\times n$ square matrix with $1$s on the leading diagonal and $0$s everywhere else.

$$0_2=\begin{pmatrix}0&0\\0&0\end{pmatrix},$$ $$I_2=\begin{pmatrix}1&0\\0&1\end{pmatrix},$$ $$I_3=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$$
Why these matrices matter

The zero and identity matrices are the matrix world's versions of the numbers $0$ and $1$: adding the zero matrix changes nothing, and (as the next lesson shows once multiplication is defined) multiplying by the identity matrix changes nothing either — $AI=IA=A$ for any square matrix $A$. Keep this analogy in mind; it makes several later results easier to remember.

Rectangular matrices and vectors

Matrices need not be square

Matrices arise with any order $m\times n$, not just square ones. A $1\times n$ matrix is a row vector; an $n\times1$ matrix is a column vector. Addition and scalar multiplication work identically for rectangular matrices, provided orders match.

Example 3 · Working with a non-square matrix
Given $A=\begin{pmatrix}1&2&-1\\0&3&4\end{pmatrix}$ (order $2\times3$) and $B=\begin{pmatrix}2&0&1\\-1&2&3\end{pmatrix}$, find $A+2B$.

$2B=\begin{pmatrix}4&0&2\\-2&4&6\end{pmatrix}$.

$$A+2B=\begin{pmatrix}1+4&2+0&-1+2\\0-2&3+4&4+6\end{pmatrix}=\begin{pmatrix}5&2&1\\-2&7&10\end{pmatrix}$$
Common mistake

Attempting to add or subtract matrices of different orders is undefined and should never be attempted — always state the order of each matrix first when checking whether an operation is valid.

Exam-style questions

Question 1 5 marks

Given $A=\begin{pmatrix}4&-2\\1&3\end{pmatrix}$ and $B=\begin{pmatrix}-1&5\\2&0\end{pmatrix}$, find $3A+2B$.

$3A=\begin{pmatrix}12&-6\\3&9\end{pmatrix}$, $2B=\begin{pmatrix}-2&10\\4&0\end{pmatrix}$ (M1, A1)

$3A+2B=\begin{pmatrix}10&4\\7&9\end{pmatrix}$ (M1, A1, A1)

Question 2 6 marks

Given that $\begin{pmatrix}2x&5\\3&x-y\end{pmatrix}=\begin{pmatrix}8&5\\3&6\end{pmatrix}$, find $x$ and $y$.

Comparing top-left entries: $2x=8 \Rightarrow x=4$ (M1, A1)

Comparing bottom-right entries: $x-y=6 \Rightarrow 4-y=6 \Rightarrow y=-2$ (M1, A1)

The top-right and bottom-left entries ($5=5$ and $3=3$) are already consistent, confirming the matrix equation holds. (A1, A1)

Question 3 5 marks

Given $A=\begin{pmatrix}1&-2&3\\4&0&-1\end{pmatrix}$ and $B=\begin{pmatrix}-2&1&0\\3&-1&2\end{pmatrix}$, find $A-2B$.

$2B=\begin{pmatrix}-4&2&0\\6&-2&4\end{pmatrix}$ (M1)

$A-2B=\begin{pmatrix}1-(-4)&-2-2&3-0\\4-6&0-(-2)&-1-4\end{pmatrix}=\begin{pmatrix}5&-4&3\\-2&2&-5\end{pmatrix}$ (M1, A1, A1, A1)

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