Identity and zero matrices | Year 11

01 / LEARN

Understand the idea.

The identity preserves vectors

The identity $I_{n}$ is square, with 1 on its main diagonal and 0 elsewhere. Its diagonal keeps each component, while its zero entries prevent mixing: multiplication by the correctly sized identity leaves a matrix unchanged.

If $A$ has order $m \times n$ , use $I_{m}$ on its left and $I_{n}$ on its right. The identities may have different orders: $I_{m} A = A = A I_{n}$ follows from the multiplication dimensions.

Zero is not a cancellation rule

A zero matrix has only zero entries, but its order still matters. Adding a same-sized zero matrix changes nothing. A defined product containing a zero factor is a zero matrix whose order follows the usual product dimensions.

The converse fails: two non-zero matrices can multiply to zero. A matrix may remove all the information carried by another matrix, so $A B = 0$ does not imply either factor is zero. Cancellation requires an appropriate inverse.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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 $A J = A$ .

Worked solution

1
Identify the order of the given matrix.
$order (A) = 2 \times 3$
There are two rows and three columns.
2
Choose the left identity.
$I = I_{2} = (\begin{array}{cccccccccc} 1 & 0 \\ 0 & 1 \end{array})$
A left identity must have two columns to meet the two rows of $A$ and preserve its row count.
3
Choose the right identity.
$J = I_{3} = (\begin{array}{cccccccccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array})$
A right identity must have three rows and preserve the three columns of $A$ .
4
Record both compatible products.
$I_{2} A = A = A I_{3}$
The identity leaves every row or column component unchanged.

Answer

$I = I_{2}$ , $J = I_{3}$ .

EXAMPLE 02

A zero product without zero factors

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

Worked solution

1
Expand the row-column products.
$A B = (\begin{array}{cccccccccc} 1 (0) + 0 (0) & 1 (0) + 0 (1) \\ 0 (0) + 0 (0) & 0 (0) + 0 (1) \end{array})$
Each row of $A$ is dotted with each column of $B$ .
2
Simplify all four entries.
$A B = (\begin{array}{cccccccccc} 0 & 0 \\ 0 & 0 \end{array}) = 0_{2 \times 2}$
Every term contains a zero factor.
3
Inspect the factors separately.
$a_{11} = 1, b_{22} = 1$
Each factor has a non-zero entry, so the zero product cannot justify declaring either factor zero.

Answer

$A B = 0_{2 \times 2}$ although neither factor is zero.

03 / YOUR TURN

Try it yourself.

Check your understanding

What does $I_{3}$ look like?

Show answer & reasoning

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

$I_{3}$ has three rows and columns, with 1 in positions $(1, 1)$ , $(2, 2)$ and $(3, 3)$ and 0 elsewhere. Those entries preserve each of the three coordinates separately.

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

The identity preserves vectors

Zero is not a cancellation rule

See the method in action.

Two correctly sized identities

Identify the order of the given matrix.

Choose the left identity.

Choose the right identity.

Record both compatible products.

A zero product without zero factors

Expand the row-column products.

Simplify all four entries.

Inspect the factors separately.

Try it yourself.

Check your understanding

Keep practising