Matrix arithmetic | Year 11

01 / LEARN

The idea, step by step

Match the positions and units

Addition combines entries in corresponding positions. Matrices need equal orders and matching labels: otherwise a row or column has no matching entry. Subtraction is also entry by entry.

Equal orders alone do not make a model sensible. Do not add kilograms to dollars. If categories are listed in different orders, rearrange them before calculating.

Scale every entry

A scalar is one number multiplying the whole matrix. It acts on every entry, including zeros and negatives. In $2 A - B$ , first double $A$ , then subtract the entire signed entry of $B$ at each position.

A negative change in stock can describe a decrease. Distinguish this from a negative physical stock count, which would need investigation. Always interpret the result rather than only displaying numbers.

A common trap

Subtracting a negative makes an answer more negative.

Instead: For instance, $4 - (- 2) = 6$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Add signed entries

Find $A + B$ for $A = (\begin{array}{cccccccccc} 2 & - 1 \\ 3 & 4 \end{array})$ and $B = (\begin{array}{cccccccccc} 5 & 2 \\ - 2 & 1 \end{array})$ .

Add corresponding entries: $2 + 5$ , $- 1 + 2$ , $3 + (- 2)$ , $4 + 1$ .
Keep their row/column positions.

A + B = (\begin{array}{cccccccccc} 7 & 1 \\ 1 & 5 \end{array})

Example 2

Interpret stock change

Opening stock is $(\begin{array}{cccccccccc} 18 & 25 \\ 12 & 30 \end{array})$ and closing stock $(\begin{array}{cccccccccc} 14 & 29 \\ 17 & 21 \end{array})$ . Find closing minus opening.

Subtract opening from closing at each position.
The differences are $- 4, 4, 5, - 9$ .

(\begin{array}{cccccccccc} - 4 & 4 \\ 5 & - 9 \end{array})

; the final category decreased by 9.

Check your understanding

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

Show answer

$(\begin{array}{cccccccccc} - 6 & 2 \end{array})$ .

Multiply both entries by −2.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Find $A - B$ for $A = (\begin{array}{cccccccccc} 2 & - 1 \\ 3 & 4 \end{array})$ and $B = (\begin{array}{cccccccccc} 5 & 2 \\ - 2 & 1 \end{array})$ .
2 marks
Find $2 A - B$ for $A = (\begin{array}{cccccccccc} 2 & - 1 \\ 3 & 4 \end{array})$ and $B = (\begin{array}{cccccccccc} 5 & 2 \\ - 2 & 1 \end{array})$ .
2 marks
Calculate $3 (\begin{array}{cccccccccc} 1 & 0 \\ - 2 & 4 \end{array})$ .
2 marks
Why cannot a $2 \times 3$ matrix be added to a $3 \times 2$ matrix?
2 marks
Find $X$ if $A + X = B$ , $A = (\begin{array}{cccccccccc} 1 & 3 \\ 2 & 4 \end{array})$ and $B = (\begin{array}{cccccccccc} 6 & 5 \\ 1 & 9 \end{array})$ .
2 marks
Should a ticket-count matrix be added to a same-order price matrix to find revenue?
2 marks

Read the worked solutions

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

Find $A - B$ for $A = (\begin{array}{cccccccccc} 2 & - 1 \\ 3 & 4 \end{array})$ and $B = (\begin{array}{cccccccccc} 5 & 2 \\ - 2 & 1 \end{array})$ .
2 marks
Subtract matching entries.
$3 - (- 2) = 5$ .
$(\begin{array}{cccccccccc} - 3 & - 3 \\ 5 & 3 \end{array})$ .
Find $2 A - B$ for $A = (\begin{array}{cccccccccc} 2 & - 1 \\ 3 & 4 \end{array})$ and $B = (\begin{array}{cccccccccc} 5 & 2 \\ - 2 & 1 \end{array})$ .
2 marks
$2 A = (\begin{array}{cccccccccc} 4 & - 2 \\ 6 & 8 \end{array})$ .
Subtract $B$ entry by entry.
$(\begin{array}{cccccccccc} - 1 & - 4 \\ 8 & 7 \end{array})$ .
Calculate $3 (\begin{array}{cccccccccc} 1 & 0 \\ - 2 & 4 \end{array})$ .
2 marks
Multiply every entry by 3.
The zero remains zero.
$(\begin{array}{cccccccccc} 3 & 0 \\ - 6 & 12 \end{array})$ .
Why cannot a $2 \times 3$ matrix be added to a $3 \times 2$ matrix?
2 marks
The rows and columns do not match.
Addition requires corresponding positions.
Their orders differ.
Find $X$ if $A + X = B$ , $A = (\begin{array}{cccccccccc} 1 & 3 \\ 2 & 4 \end{array})$ and $B = (\begin{array}{cccccccccc} 6 & 5 \\ 1 & 9 \end{array})$ .
2 marks
Use $X = B - A$ .
Subtract matching entries.
$X = (\begin{array}{cccccccccc} 5 & 2 \\ - 1 & 5 \end{array})$ .
Should a ticket-count matrix be added to a same-order price matrix to find revenue?
2 marks
Counts and prices have different units.
Multiply appropriate counts by prices, then total.
No.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: matrices and networks. Original learning resources follow the approved Ascent Library chapter map.

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