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Scalar multiplication means multiplying every entry of a matrix by the same ordinary number, called a scalar. Notes, worked examples and 8 questions answered.
Scalar Multiplication is part of Multiplication of matrices in GCSE AQA Higher Further Maths.
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Scalar multiplication means multiplying every entry of a matrix by the same ordinary number, called a scalar. Notes, worked examples and 8 questions answered.
Use this lesson to stay inside Multiplication of matrices while connecting the detail back to the wider Matrix Transformations topic.
Read the core explanations from Scalar Multiplication before testing yourself from memory.
You already know how to multiply a number by another number. Scalar multiplication takes that same idea and applies it to a whole matrix at once — every single entry gets multiplied by the same number.
That single number doing the multiplying is called a scalar. It is not a matrix — it is an ordinary number like $3$, $-2$, or $\frac{1}{2}$. The rule is refreshingly direct: multiply the scalar by every entry in the matrix, one at a time.
Notice that the order of the matrix stays the same. A $2 \times 2$ matrix multiplied by a scalar is still a $2 \times 2$ matrix. Nothing about the structure changes — only the values inside.
Here is the general rule written out clearly. If $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, then multiplying by scalar $k$ gives:
The same rule extends to matrices of any size. A $2 \times 3$ matrix has six entries — multiply the scalar by all six. A $3 \times 3$ matrix has nine entries — multiply the scalar by all nine. The process never changes.
Theory is clearest when you see it in action. The first example uses a positive integer scalar, which is the most common case you will meet in exam questions.
Worked example: Scalar multiplication — positive integer scalar
Calculate $3 \begin{pmatrix} 2 & -1 \\ 4 & 0 \end{pmatrix}$.
Answer: $\begin{pmatrix} 6 & -3 \\ 12 & 0 \end{pmatrix}$
Now look at what happens when the scalar is negative. The arithmetic is the same — but you must track the sign of every entry carefully.
Worked example: Scalar multiplication — negative scalar
Calculate $-2 \begin{pmatrix} 5 & -3 \\ 0 & 7 \end{pmatrix}$.
Answer: $\begin{pmatrix} -10 & 6 \\ 0 & -14 \end{pmatrix}$
Scalars do not have to be whole numbers. You might be asked to multiply a matrix by $\frac{1}{2}$, $0.5$, or even $\frac{3}{4}$. The method is identical — multiply every entry by the scalar.
Worked example: Scalar multiplication — fractional scalar
Calculate $\frac{1}{2} \begin{pmatrix} 6 & -4 \\ 10 & 2 \end{pmatrix}$.
Answer: $\begin{pmatrix} 3 & -2 \\ 5 & 1 \end{pmatrix}$
This example also shows something worth noting: multiplying by a scalar between 0 and 1 makes the entries smaller in magnitude, while multiplying by a scalar greater than 1 makes them larger. Multiplying by a negative scalar flips all the signs.
The rule scales up without any change in method. Whether the matrix is $2 \times 3$, $3 \times 2$, or $3 \times 3$, you still multiply every entry by the scalar.
Worked example: Scalar multiplication — 2×3 matrix
Calculate $-3 \begin{pmatrix} 1 & 0 & -2 \\ 4 & -1 & 3 \end{pmatrix}$.
Answer: $\begin{pmatrix} -3 & 0 & 6 \\ -12 & 3 & -9 \end{pmatrix}$
Before you practise independently, here is a compact summary of everything you need for scalar multiplication questions.
Scalar multiplication questions are reliable mark-earners — the method is straightforward, but there are a handful of traps that appear repeatedly in exam papers. Knowing them in advance puts you in control.
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-14.
Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
Scalar multiplication means multiplying every entry of a matrix by the same ordinary number, called a scalar. If k is a scalar and A is a matrix, kA is formed by multiplying every entry of A by k, and the size (order) of the matrix does not change.
Multiply each entry by the number and put each product back in its original position. For example, 3 times the matrix with rows (2, −1) and (4, 0) gives rows (6, −3) and (12, 0); the zero entry still needs to be written.
Multiply every entry by the negative scalar, tracking signs carefully, because every entry changes sign. For example, −2 times the matrix with rows (5, −3) and (0, 7) gives rows (−10, 6) and (0, −14), since (−2) × (−3) = +6.
Multiply every entry by the fraction. Multiplying by 1/2 halves every entry, so 1/2 times the matrix with rows (6, −4) and (10, 2) gives rows (3, −2) and (5, 1). A scalar between 0 and 1 makes entries smaller in magnitude.
Scalar multiplication never changes the order of a matrix. A 2 × 3 matrix stays 2 × 3: for example, −3 times the matrix with rows (1, 0, −2) and (4, −1, 3) gives rows (−3, 0, 6) and (−12, 3, −9), six entries in and six out.
A scalar is an ordinary number, such as 3, −2 or ½, that multiplies a matrix; it is not a matrix itself. In scalar multiplication, every entry of the matrix is multiplied by that same number.
A matrix is multiplied by 3 by multiplying every entry by 3. For the matrix with rows (2, −1) and (4, 0), this gives rows (6, −3) and (12, 0), including writing 3 × 0 = 0 for the zero entry.
Multiplying a matrix by a scalar between 0 and 1 makes the entries smaller in magnitude, while a scalar greater than 1 makes them larger and a negative scalar flips all the signs. For example, ½ times the matrix with rows (6, −4) and (10, 2) gives rows (3, −2) and (5, 1).