Identity and zero matrices | Year 11

01 / LEARN

The idea, step by step

The identity preserves vectors

The square identity matrix $I_{n}$ has ones on its main diagonal and zeros elsewhere. Multiplying a compatible matrix by an identity leaves it unchanged: $I_{m} A = A = A I_{n}$ when A is $m \times n$ .

The identity is not a matrix filled with ones. Its off-diagonal zero entries prevent components from mixing.

Zero is not a cancellation rule

A zero matrix has every entry zero. Adding the compatible zero matrix changes nothing, and any defined product with a zero matrix is zero.

Unlike real numbers, non-zero matrices can have a zero product. Therefore $A B = 0$ does not by itself imply $A = 0$ or $B = 0$ . Invertibility, introduced later, is the extra condition that permits cancellation.

A common trap

Cancelling a matrix from a product without checking invertibility.

Instead: Zero products and equal products need extra care; a non-zero matrix need not be invertible.

02 / WORKED EXAMPLES

See the thinking

Example 1

Two correctly sized identities

For $A = (\begin{array}{cccccccccc} 2 & 3 & 1 \\ - 1 & 0 & 4 \end{array})$ , state the identities in $I A = A$ and $A J = A$ .

A has order $2 \times 3$ .
The left identity must be $I_{2}$ ; the right identity must be $I_{3}$ .

I = I_{2}

J = I_{3}

Example 2

A zero product without zero factors

Find $A B$ for $A = (\begin{array}{cccccccccc} 1 & 0 \\ 0 & 0 \end{array})$ , $B = (\begin{array}{cccccccccc} 0 & 0 \\ 0 & 1 \end{array})$ .

Every row-column dot product is zero.
A and B each contain a 1, so neither is a zero matrix.

A B = 0_{2 \times 2}

although neither factor is zero.

Check your understanding

What does $I_{3}$ look like?

Show answer

$(\begin{array}{cccccccccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array})$ .

Only the diagonal entries are 1.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Find $I_{2} (\begin{array}{cccccccccc} 4 & - 2 \\ 1 & 5 \end{array})$ .
2 marks
State the orders of identities needed on the left and right of a $4 \times 2$ matrix.
2 marks
Calculate $(\begin{array}{cccccccccc} 1 & 2 \\ 3 & 4 \end{array}) 0_{2 \times 3}$ .
2 marks
If $A + X = A$ for a $2 \times 3$ matrix A, find X.
2 marks
Let $P = (\begin{array}{cccccccccc} 0 & 1 \\ 0 & 0 \end{array})$ . Find $P^{2}$ and say whether $P = 0$ .
2 marks
Explain why a matrix filled with ones is not $I_{2}$ .
2 marks

Read the worked solutions

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

Find $I_{2} (\begin{array}{cccccccccc} 4 & - 2 \\ 1 & 5 \end{array})$ .
2 marks
The identity leaves every column unchanged.
$(\begin{array}{cccccccccc} 4 & - 2 \\ 1 & 5 \end{array})$ .
State the orders of identities needed on the left and right of a $4 \times 2$ matrix.
2 marks
Use $I_{4} A = A = A I_{2}$ .
Left: $4 \times 4$ ; right: $2 \times 2$ .
Calculate $(\begin{array}{cccccccccc} 1 & 2 \\ 3 & 4 \end{array}) 0_{2 \times 3}$ .
2 marks
The product exists and has order $2 \times 3$ .
Each row-column sum contains only zero products.
$0_{2 \times 3}$ .
If $A + X = A$ for a $2 \times 3$ matrix A, find X.
2 marks
Subtract A entry by entry.
$X = 0_{2 \times 3}$ .
Let $P = (\begin{array}{cccccccccc} 0 & 1 \\ 0 & 0 \end{array})$ . Find $P^{2}$ and say whether $P = 0$ .
2 marks
Direct multiplication gives four zero entries.
P has a non-zero top-right entry.
$P^{2} = 0$ , but $P \neq 0$ .
Explain why a matrix filled with ones is not $I_{2}$ .
2 marks
For example it sends $(1, 0)$ to $(1, 1)$ , not $(1, 0)$ .
It mixes the input components.

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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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