Singular matrices | Year 11

01 / LEARN

Understand the idea.

What singular means

A square matrix is singular when its determinant is zero. For a non-zero singular $2 \times 2$ matrix, its dependent columns force every output onto one line through the origin. The zero matrix instead sends every input to the origin.

Collapse loses information: distinct inputs can have the same output. An inverse could not know which input to return, so no singular square matrix has an inverse that recovers every original vector.

A singular system is not automatically inconsistent

Test whether $b$ is an attainable output. A nonzero singular $2 \times 2$ matrix has a line image; the zero matrix has image ${0}$ . A consistent singular system has infinitely many solutions; otherwise it has none.

Test consistency using the original equations. If one left side is twice another, its right side must also double. A zero determinant rules out a unique solution, but alone does not distinguish inconsistency from infinitely many solutions.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A collapsed plane

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

Worked solution

1
Multiply by the general input vector.
$A (\begin{array}{cccccccccc} x \\ y \end{array}) = (\begin{array}{cccccccccc} x + 2 y \\ 2 x + 4 y \end{array})$
A general input describes the entire image, rather than just selected test points.
2
Factor the dependence between output coordinates.
$s = x + 2 y, A (\begin{array}{cccccccccc} x \\ y \end{array}) = (\begin{array}{cccccccccc} s \\ 2 s \end{array})$
The second output coordinate is twice the first for every input.
3
Identify the output line.
$X = s, Y = 2 s ⟹ Y = 2 X$
Capital letters distinguish output coordinates from the original input coordinates.
4
Check that the whole line is reached.
$A (\begin{array}{cccccccccc} s \\ 0 \end{array}) = (\begin{array}{cccccccccc} s \\ 2 s \end{array}) (s \in ℝ)$
Choosing this input produces every real value of the line parameter.

Answer

$A {(x, y)}^{T} = {(x + 2 y, 2 x + 4 y)}^{T}$ , on $Y = 2 X$ .

EXAMPLE 02

Two right-hand sides

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

Worked solution

1
Double the first equation.
$2 (x + 2 y) = 2 (3) ⟹ 2 x + 4 y = 6$
Any solution of the first equation must satisfy its doubled version.
2
Parametrise the first system.
$y = t, x = 3 - 2 t, t \in ℝ$
The second equation repeats the first, leaving one free real parameter.
3
Verify the family.
$2 (3 - 2 t) + 4 t = 6$
Every parameter value satisfies the second equation as well as the first.
4
Compare the second system.
$2 x + 4 y = 6 and 2 x + 4 y = 7$
The same left side cannot equal two different constants, so this system has no solution.

Answer

First: infinitely many solutions $(3 - 2 t, t)$ . Second: no solution.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

No.

No. A singular square matrix sends some distinct inputs to the same output, so a two-sided inverse cannot recover both. Equivalently, its zero determinant cannot multiply with another determinant to give $det I = 1$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

What singular means

A singular system is not automatically inconsistent

See the method in action.

A collapsed plane

Multiply by the general input vector.

Factor the dependence between output coordinates.

Identify the output line.

Check that the whole line is reached.

Two right-hand sides

Double the first equation.

Parametrise the first system.

Verify the family.

Compare the second system.

Try it yourself.

Check your understanding

Keep practising