Addition/subtraction/scalar multiplication | Year 11

01 / LEARN

The idea, step by step

Combine matching positions

Matrix addition and subtraction are entry-by-entry operations. The matrices must have the same order: a $2 \times 3$ matrix cannot be added to a $3 \times 2$ matrix even though both have six entries.

For subtraction, subtract each entry of the second matrix. Negative entries need brackets so that a double negative is not lost.

Scale and rearrange

Multiplication by a scalar multiplies every entry. The scalar may be negative or fractional. Distribute it before adding matrices, or use a valid common factor to simplify the arithmetic.

An equation such as $2 X + A = B$ can be rearranged as $X = \frac{1}{2} (B - A)$ , because all matrices have the same order and the operations are addition and scalar multiplication.

A common trap

Multiplying only one entry by a scalar.

Instead: The scalar applies to the whole matrix, so every entry must be multiplied.

02 / WORKED EXAMPLES

See the thinking

Example 1

Subtract negative entries

Find $(\begin{array}{cccccccccc} 3 & - 2 \\ 1 & 4 \end{array}) - (\begin{array}{cccccccccc} - 1 & 5 \\ 2 & - 3 \end{array})$ .

Compute $3 - (- 1) = 4$ and $- 2 - 5 = - 7$ .
Compute $1 - 2 = - 1$ and $4 - (- 3) = 7$ .

(\begin{array}{cccccccccc} 4 & - 7 \\ - 1 & 7 \end{array})

Example 2

Solve for an unknown matrix

Solve $2 X + (\begin{array}{cccccccccc} 1 & 3 \\ - 2 & 0 \end{array}) = (\begin{array}{cccccccccc} 5 & 1 \\ 4 & 6 \end{array})$ .

Subtract the known matrix: $2 X = (\begin{array}{cccccccccc} 4 & - 2 \\ 6 & 6 \end{array})$ .
Divide every entry by 2.

X = (\begin{array}{cccccccccc} 2 & - 1 \\ 3 & 3 \end{array})

Check your understanding

What order condition is needed to calculate $A - B$ ?

Show answer

The orders must be equal.

Each entry in one matrix must have a matching position in the other.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Calculate $(\begin{array}{cccccccccc} 1 & 4 \end{array}) + (\begin{array}{cccccccccc} 3 & - 2 \end{array})$ .
2 marks
Calculate $- 3 (\begin{array}{cccccccccc} 2 & - 1 \\ 0 & 4 \end{array})$ .
2 marks
Calculate $(\begin{array}{cccccccccc} 5 & 2 \\ - 1 & 7 \end{array}) - (\begin{array}{cccccccccc} 2 & - 3 \\ 4 & 1 \end{array})$ .
2 marks
Find $X$ if $X + (\begin{array}{cccccccccc} 2 & 0 \\ 1 & 3 \end{array}) = (\begin{array}{cccccccccc} 5 & 4 \\ - 2 & 8 \end{array})$ .
2 marks
Find $\frac{1}{2} A + 2 B$ for $A = (\begin{array}{cccccccccc} 4 & 2 \end{array})$ , $B = (\begin{array}{cccccccccc} - 1 & 3 \end{array})$ .
2 marks
Explain why a $2 \times 2$ matrix and a $2 \times 3$ matrix cannot be added.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Calculate $(\begin{array}{cccccccccc} 1 & 4 \end{array}) + (\begin{array}{cccccccccc} 3 & - 2 \end{array})$ .
2 marks
Add corresponding entries: $1 + 3 = 4$ , $4 - 2 = 2$ .
$(\begin{array}{cccccccccc} 4 & 2 \end{array})$ .
Calculate $- 3 (\begin{array}{cccccccccc} 2 & - 1 \\ 0 & 4 \end{array})$ .
2 marks
Multiply all four entries by -3.
$(\begin{array}{cccccccccc} - 6 & 3 \\ 0 & - 12 \end{array})$ .
Calculate $(\begin{array}{cccccccccc} 5 & 2 \\ - 1 & 7 \end{array}) - (\begin{array}{cccccccccc} 2 & - 3 \\ 4 & 1 \end{array})$ .
2 marks
Subtract entries in matching positions.
In particular, $2 - (- 3) = 5$ and $- 1 - 4 = - 5$ .
$(\begin{array}{cccccccccc} 3 & 5 \\ - 5 & 6 \end{array})$ .
Find $X$ if $X + (\begin{array}{cccccccccc} 2 & 0 \\ 1 & 3 \end{array}) = (\begin{array}{cccccccccc} 5 & 4 \\ - 2 & 8 \end{array})$ .
2 marks
Subtract the known matrix from the right side.
$X = (\begin{array}{cccccccccc} 3 & 4 \\ - 3 & 5 \end{array})$ .
Find $\frac{1}{2} A + 2 B$ for $A = (\begin{array}{cccccccccc} 4 & 2 \end{array})$ , $B = (\begin{array}{cccccccccc} - 1 & 3 \end{array})$ .
2 marks
Half of $A$ is $(2, 1)$ .
Twice $B$ is $(- 2, 6)$ ; add to obtain $(0, 7)$ .
$(\begin{array}{cccccccccc} 0 & 7 \end{array})$ .
Explain why a $2 \times 2$ matrix and a $2 \times 3$ matrix cannot be added.
2 marks
The first matrix has no third-column entries to match the second matrix.
The orders differ.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: matrices and transformations; original lessons follow the approved school-sized chapter structure.

Official curriculum source ↗

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