Singular matrices | Year 11

01 / LEARN

The idea, step by step

What singular means

A square matrix is singular when its determinant is zero. Its columns are dependent: for a non-zero singular $2 \times 2$ matrix, every output lies on one line through the origin. The zero matrix sends everything to the origin.

Different input vectors can produce the same output. Once that information has been lost, no matrix can reverse the transformation for every vector.

A singular system is not automatically inconsistent

In $A X = b$ , a singular A may give no solutions or infinitely many solutions. If b is outside the output line, there is no solution. If b lies on it, there are many input vectors with that output.

Use the original equations to decide which case applies. A zero determinant rules out a unique solution; it does not decide between none and infinitely many.

A common trap

Saying every singular system has no solution.

Instead: Check whether the right-hand side is consistent with the dependence between rows.

02 / WORKED EXAMPLES

See the thinking

Example 1

A collapsed plane

For $A = (\begin{array}{cccccccccc} 1 & 2 \\ 2 & 4 \end{array})$ , find the image of $(x, y)$ .

Multiply to obtain $(x + 2 y, 2 x + 4 y)$ .
Writing $s = x + 2 y$ , every output is $(s, 2 s)$ .
Thus the output lies on $Y = 2 X$ .

A {(x, y)}^{T} = {(x + 2 y, 2 x + 4 y)}^{T}

, on

Y = 2 X

Example 2

Two right-hand sides

Compare $x + 2 y = 3, 2 x + 4 y = 6$ with $x + 2 y = 3, 2 x + 4 y = 7$ .

In the first system the second equation is twice the first.
In the second, doubling the first gives 6, contradicting 7.

First: infinitely many solutions

(3 - 2 t, t)

. Second: no solution.

Check your understanding

Can a singular square matrix have a two-sided inverse?

Show answer

No.

Its transformation loses information, and its determinant is zero.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Is $(\begin{array}{cccccccccc} 2 & 6 \\ 1 & 3 \end{array})$ singular?
2 marks
Find $k$ so that $(\begin{array}{cccccccccc} 2 & k \\ 4 & 6 \end{array})$ is singular.
2 marks
Solve $x + y = 4$ , $2 x + 2 y = 8$ using a parameter.
2 marks
Explain why $x + y = 4$ , $2 x + 2 y = 9$ is inconsistent.
2 marks
Find a non-zero vector sent to zero by $(\begin{array}{cccccccccc} 1 & - 2 \\ 2 & - 4 \end{array})$ .
2 marks
The matrix $(\begin{array}{cccccccccc} 3 & 0 \\ 0 & 0 \end{array})$ acts on the plane. Describe all its possible outputs.
2 marks

Read the worked solutions

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

Is $(\begin{array}{cccccccccc} 2 & 6 \\ 1 & 3 \end{array})$ singular?
2 marks
The determinant is $6 - 6 = 0$ .
Yes.
Find $k$ so that $(\begin{array}{cccccccccc} 2 & k \\ 4 & 6 \end{array})$ is singular.
2 marks
Set $12 - 4 k = 0$ .
$k = 3$ .
Solve $x + y = 4$ , $2 x + 2 y = 8$ using a parameter.
2 marks
The second equation repeats the first.
Choose $y = t$ and obtain $x = 4 - t$ .
$(x, y) = (4 - t, t)$ for any real t.
Explain why $x + y = 4$ , $2 x + 2 y = 9$ is inconsistent.
2 marks
No pair can satisfy both statements.
Doubling the first equation gives 8, not 9.
Find a non-zero vector sent to zero by $(\begin{array}{cccccccccc} 1 & - 2 \\ 2 & - 4 \end{array})$ .
2 marks
$2 - 2 (1) = 0$ and $2 (2) - 4 (1) = 0$ .
For example $(2, 1)$ .
The matrix $(\begin{array}{cccccccccc} 3 & 0 \\ 0 & 0 \end{array})$ acts on the plane. Describe all its possible outputs.
2 marks
The output is $(3 x, 0)$ .
Every real x-coordinate is possible, but the y-coordinate is always zero.
All points on the x-axis.

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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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