Addition, subtraction and multiplication by a real number
Master matrix addition, subtraction and multiplication by a real number for Year 11 Specialist Mathematics in Queensland (QCAA). These are the first operations of matrix algebra: combining matrices of the same order entry by entry, and scaling every entry by a real number.
You will learn when matrices can be added, how a scalar multiplies every entry, the roles of the zero and inverse matrices, and how to use the commutative, associative and distributive properties to solve matrix equations — the groundwork for matrix multiplication and inverses later in the course.
Theory
Matrix addition, subtraction and multiplication by a real number are the first operations of matrix algebra in Year 11 Specialist Mathematics (QCAA, Queensland). You add or subtract matrices of the same order entry by entry, and a scalar (real number) multiplies every entry. This page shows the rules, the zero and inverse matrices, the algebra properties, and how to solve simple matrix equations.
A matrix is a rectangular array of numbers set in brackets. Its order (or size) is written rows×columns, so \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is a \(2\times 2\) matrix. The number in row \(i\), column \(j\) is the entry \(a_{ij}\).
To add or subtract two matrices they must have exactly the same order (this is called being conformable for addition). You then add or subtract the entries in matching positions; the answer has the same order.
Multiplication by a real number \(k\) (a scalar) multiplies every entry by \(k\). Doubling a matrix doubles each entry; multiplying by \(-1\) negates each entry.
The zero matrix \(O\) (all entries \(0\)) is the additive identity: \(A+O=A\). The matrix \(-A\) is the additive inverse of \(A\): \(A+(-A)=O\). Matrix addition is commutative (\(A+B=B+A\)) and associative (\((A+B)+C=A+(B+C)\)), and scalar multiplication is distributive over addition (\(k(A+B)=kA+kB\)).
For matrices \(A=\begin{pmatrix} a_1 & a_2 \\ a_3 & a_4 \end{pmatrix}\) and \(B=\begin{pmatrix} b_1 & b_2 \\ b_3 & b_4 \end{pmatrix}\) of the same order, and a real number \(k\):
To solve an additive matrix equation, make \(X\) the subject exactly as in ordinary algebra, then combine entry by entry:
How to combine matrices
- Check the order: for \(A+B\) or \(A-B\), confirm both matrices have the same order — if not, the sum or difference is undefined.
- Scale first if needed: for a combination such as \(2A-3B\), multiply each matrix by its scalar (every entry) before combining.
- Combine entry by entry: add or subtract the numbers in matching positions, keeping careful track of signs.
- To solve for \(X\): rearrange the equation to make \(X\) the subject (e.g. \(X=B-A\)), then do the matrix arithmetic; divide every entry when a scalar coefficient remains.
Both matrices are \(2\times 2\), so add the entries in matching positions:
| \(A+B\) | \(=\) | \(\begin{pmatrix}3 & -4 \\ 6 & 2\end{pmatrix}+\begin{pmatrix}1 & 5 \\ -2 & 7\end{pmatrix}\) |
| \(=\) | \(\begin{pmatrix}3+1 & -4+5 \\ 6-2 & 2+7\end{pmatrix}\) | |
| \(=\) | \(\begin{pmatrix}4 & 1 \\ 4 & 9\end{pmatrix}\) |
\(A+B=\begin{pmatrix} 4 & 1 \\ 4 & 9 \end{pmatrix}\).
Multiply each matrix by its scalar first, then subtract entry by entry:
| \(2A\) | \(=\) | \(\begin{pmatrix}2\times 4 & 2\times 0 \\ 2\times(-1) & 2\times 3\end{pmatrix}\) |
| \(=\) | \(\begin{pmatrix}8 & 0 \\ -2 & 6\end{pmatrix}\) | |
| \(3B\) | \(=\) | \(\begin{pmatrix}3\times 2 & 3\times 5 \\ 3\times 3 & 3\times(-2)\end{pmatrix}\) |
| \(=\) | \(\begin{pmatrix}6 & 15 \\ 9 & -6\end{pmatrix}\) | |
| \(2A-3B\) | \(=\) | \(\begin{pmatrix}8-6 & 0-15 \\ -2-9 & 6-(-6)\end{pmatrix}\) |
| \(=\) | \(\begin{pmatrix}2 & -15 \\ -11 & 12\end{pmatrix}\) |
\(2A-3B=\begin{pmatrix} 2 & -15 \\ -11 & 12 \end{pmatrix}\).
Make \(X\) the subject by subtracting \(A\) from both sides, then combine:
| \(X\) | \(=\) | \(B-A\) |
| \(=\) | \(\begin{pmatrix}5 & 4 \\ 0 & 7\end{pmatrix}-\begin{pmatrix}2 & 6 \\ -3 & 1\end{pmatrix}\) | |
| \(=\) | \(\begin{pmatrix}5-2 & 4-6 \\ 0-(-3) & 7-1\end{pmatrix}\) | |
| \(=\) | \(\begin{pmatrix}3 & -2 \\ 3 & 6\end{pmatrix}\) |
\(X=\begin{pmatrix} 3 & -2 \\ 3 & 6 \end{pmatrix}\).
Add the two days entry by entry for the total:
| \(S+N\) | \(=\) | \(\begin{pmatrix}40+36 & 25+29 \\ 30+22 & 18+20\end{pmatrix}\) |
| \(=\) | \(\begin{pmatrix}76 & 54 \\ 52 & 38\end{pmatrix}\) |
Double every entry of \(S\) for the promotion:
| \(2S\) | \(=\) | \(\begin{pmatrix}2\times 40 & 2\times 25 \\ 2\times 30 & 2\times 18\end{pmatrix}\) |
| \(=\) | \(\begin{pmatrix}80 & 50 \\ 60 & 36\end{pmatrix}\) |
\(S+N=\begin{pmatrix} 76 & 54 \\ 52 & 38 \end{pmatrix}\) and \(2S=\begin{pmatrix} 80 & 50 \\ 60 & 36 \end{pmatrix}\).
Common pitfalls
Frequently asked questions
How do you add two matrices?
They must be the same order. Then add the entries in matching positions; the answer keeps that order.
Can you add matrices of different sizes?
No. Matrix addition and subtraction are only defined for matrices of equal order (same number of rows and columns).
What does multiplying a matrix by a number do?
Multiplying by a real number \(k\) (a scalar) multiplies every entry by \(k\). For example \(2A\) doubles each entry of \(A\).
What is the zero matrix?
The zero matrix \(O\) has every entry \(0\). It is the additive identity, so \(A+O=A\) for any matrix \(A\) of the same order.
Is matrix addition commutative?
Yes. \(A+B=B+A\) and \((A+B)+C=A+(B+C)\), so matrix addition is both commutative and associative — unlike matrix multiplication.
How do you solve \(A+X=B\) for a matrix \(X\)?
Rearrange to \(X=B-A\), then subtract \(A\) from \(B\) entry by entry.