IGCSEMathsEdexcelMaths BRevision NotesMatricesProperties of MatricesAdding Matrices & Multiplying by a Scalar

Adding Matrices & Multiplying by a Scalar (Edexcel IGCSE Maths B): Revision Note

Exam code: 4MB1

Written by: Roger B

Reviewed by: Jamie Wood

Updated on 14 January 2026

Basic operations with matrices

How do I add or subtract two matrices?

To add or subtract two matrices, they need to be of the same order (i.e. the same number of rows and columns)
If they are of the same order, then
- to add two matrices, just add the corresponding elements
  - $(\begin{matrix} 2 & - 3 \\ 5 & 1 \end{matrix}) + (\begin{matrix} - 1 & 7 \\ - 4 & 6 \end{matrix}) = (\begin{matrix} 2 + (- 1) & - 3 + 7 \\ 5 + (- 4) & 1 + 6 \end{matrix}) = (\begin{matrix} 1 & 4 \\ 1 & 7 \end{matrix})$
- to subtract two matrices just subtract the corresponding elements
  - $(\begin{matrix} 2 & - 3 \\ 5 & 1 \end{matrix}) - (\begin{matrix} - 1 & 7 \\ - 4 & 6 \end{matrix}) = (\begin{matrix} 2 - (- 1) & - 3 - 7 \\ 5 - (- 4) & 1 - 6 \end{matrix}) = (\begin{matrix} 3 & - 10 \\ 9 & - 5 \end{matrix})$

How do I multiply a matrix by a scalar?

To multiply any matrix by a scalar (a number), multiply each element by that scalar
- If $A = (\begin{matrix} 5 & 2 \\ 0 & 4 \end{matrix})$ then $2 A = 2 (\begin{matrix} 5 & 2 \\ 0 & 4 \end{matrix}) = (\begin{matrix} 2 \times 5 & 2 \times 2 \\ 2 \times 0 & 2 \times 4 \end{matrix}) = (\begin{matrix} 10 & 4 \\ 0 & 8 \end{matrix})$
Lower case letters often refer to scalar multiples
- $k A$ is the matrix $A$ multiplied by the scalar $k$

Worked Example

$A = (\begin{matrix} - 2 & 2 \\ - 2 & - 5 \end{matrix}) B = (\begin{matrix} - 4 & 1 \\ 3 & - 4 \end{matrix})$

(a) Calculate $A + B$

Answer:

Add the corresponding elements of the two matrices

$\begin{array}{rcl} A + B & = & (\begin{matrix} - 2 & 2 \\ - 2 & - 5 \end{matrix}) + (\begin{matrix} - 4 & 1 \\ 3 & - 4 \end{matrix}) \\ = & (\begin{matrix} - 2 + (- 4) & 2 + 1 \\ - 2 + 3 & - 5 + (- 4) \end{matrix}) \\ = & (\begin{matrix} - 6 & 3 \\ 1 & - 9 \end{matrix}) \end{array}$

$\begin{array}{rcl} A + B & = & (\begin{matrix} - 6 & 3 \\ 1 & - 9 \end{matrix}) \end{array}$

Given that $2 A - n B = (\begin{matrix} 8 & 1 \\ - 13 & 2 \end{matrix})$

(b) find the value of $n$

Answer:

Multiply all elements of $A$ by 2 to find $2 A$

$\begin{array}{rcl} 2 A & = & 2 (\begin{matrix} - 2 & 2 \\ - 2 & - 5 \end{matrix}) \\ = & (\begin{matrix} 2 \times (- 2) & 2 \times 2 \\ 2 \times (- 2) & 2 \times (- 5) \end{matrix}) \\ = & (\begin{matrix} - 4 & 4 \\ - 4 & - 10 \end{matrix}) \end{array}$

Multiply all elements of $B$ by $n$ to find $n B$

$\begin{array}{rcl} n B & = & n (\begin{matrix} - 4 & 1 \\ 3 & - 4 \end{matrix}) \\ = & (\begin{matrix} - 4 n & n \\ 3 n & - 4 n \end{matrix}) \end{array}$

Subtract the corresponding elements to find $2 A - n B$

$\begin{array}{rcl} 2 A - n B & = & (\begin{matrix} - 4 & 4 \\ - 4 & - 10 \end{matrix}) - (\begin{matrix} - 4 n & n \\ 3 n & - 4 n \end{matrix}) \\ = & (\begin{matrix} - 4 - (- 4 n) & 4 - n \\ - 4 - 3 n & - 10 - (- 4 n) \end{matrix}) \\ = & (\begin{matrix} 4 n - 4 & 4 - n \\ - 3 n - 4 & 4 n - 10 \end{matrix}) \end{array}$

Substitute that into the equation from the question

$\begin{array}{rcl} (\begin{matrix} 4 n - 4 & 4 - n \\ - 3 n - 4 & 4 n - 10 \end{matrix}) & = & (\begin{matrix} 8 & 1 \\ - 13 & 2 \end{matrix}) \end{array}$

For those two matrices to be equal, the correct value of $n$ must make ALL the corresponding elements equal in the two matrices

So you can choose any one pair to solve to find $n$

$\begin{array}{rcl} 4 n - 4 & = & 8 \\ 4 n & = & 12 \\ n & = & 3 \end{array}$

It is worth checking the other three pairs of elements with that value of $n$ , to make sure you haven't made a mistake

$4 - 3 = 1 ✔ - 3 (3) - 4 = - 13 ✔ 4 (3) - 10 = 2 ✔$

$n = 3$

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