Matrix arithmetic | Year 11

01 / LEARN

Understand the idea.

Match the positions and units

Matrix addition and subtraction combine corresponding entries. The orders must match so every position has a partner. Keep each calculation in its original row and column, including when an entry is zero or negative.

In a data model, matching dimensions are only the first check. The same positions must represent the same categories and compatible units. Reorder differently listed products before adding, and do not add a dollar price to a stock count.

Scale every entry

A scalar multiplies every entry of a matrix. For $2 A - B$ , first double all entries in $A$ , then subtract each complete signed entry in $B$ . Subtracting a negative entry is addition.

Interpret negative results using the quantity being calculated. Closing stock minus opening stock can be negative because it records a decrease. A negative closing stock count would instead indicate inconsistent data or a failed physical assumption.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Add signed entries

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

Worked solution

1
Check the orders.
$A : 2 \times 2, B : 2 \times 2$
Equal orders allow each position to be paired.
2
Add corresponding entries.
$A + B = (\begin{array}{cccccccccc} 2 + 5 & - 1 + 2 \\ 3 + (- 2) & 4 + 1 \end{array})$
Each sum stays in the row and column of its two inputs.
3
Evaluate the four sums.
$A + B = (\begin{array}{cccccccccc} 7 & 1 \\ 1 & 5 \end{array})$
Signed arithmetic gives the resulting entry in each position.

Answer

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

EXAMPLE 02

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.

Worked solution

1
Set the direction of change.
$Δ S = S_{closing} - S_{opening}$
Change means final value minus starting value.
2
Subtract entry by entry.
$Δ S = (\begin{array}{cccccccccc} 14 - 18 & 29 - 25 \\ 17 - 12 & 21 - 30 \end{array})$
Each closing stock count is compared with the same opening category.
3
Evaluate the differences.
$Δ S = (\begin{array}{cccccccccc} - 4 & 4 \\ 5 & - 9 \end{array})$
Negative entries show stock decreases; positive entries show increases.
4
Interpret the last category.
$Δ s_{22} = - 9$
The final row-and-column category has nine fewer items, not negative nine items in stock.

Answer

$(\begin{array}{cccccccccc} - 4 & 4 \\ 5 & - 9 \end{array})$ ; the final category decreased by 9.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

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

Distribute the scalar to both entries: $- 2 (3) = - 6$ and $- 2 (- 1) = 2$ , giving $(\begin{array}{cccccccccc} - 6 & 2 \end{array})$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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.

Official curriculum source ↗

In this lesson

Understand the idea.

Match the positions and units

Scale every entry

See the method in action.

Add signed entries

Check the orders.

Add corresponding entries.

Evaluate the four sums.

Interpret stock change

Set the direction of change.

Subtract entry by entry.

Evaluate the differences.

Interpret the last category.

Try it yourself.

Check your understanding

Keep practising