Determinant and area scale | Year 11

01 / LEARN

The idea, step by step

Area from the image of a unit square

The columns of a transformation matrix are the sides of the image of the unit square. Their determinant gives a signed area; ordinary area is non-negative, so use its absolute value.

Every region has the same area multiplier. For a triangle, first find its original area and then multiply by the absolute determinant. Do not insert another factor of one half after that.

Orientation and loss of dimension

A positive determinant preserves orientation; a negative determinant reverses it. The magnitude determines area scale, not each individual length scale. A map can stretch one direction and compress another while preserving area.

A zero determinant collapses the plane to a line or point, making every image region have zero area. An inverse, when it exists, has the reciprocal area scale factor.

A common trap

Reporting a negative physical area because the determinant is negative.

Instead: Use the absolute determinant for area and discuss orientation separately.

02 / WORKED EXAMPLES

See the thinking

Example 1

A reflected triangle

A triangle has area $7 {cm}^{2}$ . Find its image area under $(\begin{array}{cccccccccc} 2 & 1 \\ 0 & - 3 \end{array})$ .

The determinant is -6, giving area factor 6.
Multiply 7 by 6; the sign records reversal of orientation.

42 {cm}^{2}

Example 2

Area without length preservation

Describe the effects of $(\begin{array}{cccccccccc} 2 & 0 \\ 0 & \frac{1}{2} \end{array})$ .

The determinant is 1, so area is unchanged.
The first coordinate direction doubles in length and the second halves.

Areas are preserved, but general lengths and angles are not.

Check your understanding

A region of area 12 maps to area 3. Find the absolute determinant.

Show answer

$\frac{1}{4}$ .

Image area divided by original area is 3/12.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Find the area factor of $(\begin{array}{cccccccccc} 3 & 0 \\ 0 & 2 \end{array})$ .
2 marks
An area-5 region is mapped by $(\begin{array}{cccccccccc} 1 & 2 \\ 3 & 4 \end{array})$ . Find its image area and orientation change.
2 marks
Triangle $(0, 0), (2, 0), (0, 3)$ is dilated by factor 4. Find its image area.
2 marks
If $det A = - 5$ , state the determinant and area factor of $A^{- 1}$ .
2 marks
Find the area effect of $(\begin{array}{cccccccccc} 1 & 2 \\ 2 & 4 \end{array})$ .
2 marks
The diagonal matrix $(\begin{array}{cccccccccc} k & 0 \\ 0 & 3 \end{array})$ sends an area-4 region to area 24. Find all real k.
2 marks

Read the worked solutions

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

Find the area factor of $(\begin{array}{cccccccccc} 3 & 0 \\ 0 & 2 \end{array})$ .
2 marks
The determinant is 6.
$6$ .
An area-5 region is mapped by $(\begin{array}{cccccccccc} 1 & 2 \\ 3 & 4 \end{array})$ . Find its image area and orientation change.
2 marks
The determinant is -2.
The area factor is 2.
Area 10; orientation reverses.
Triangle $(0, 0), (2, 0), (0, 3)$ is dilated by factor 4. Find its image area.
2 marks
Its original area is $\frac{1}{2} (2) (3) = 3$ .
The determinant of 4I is 16, so the image area is 48.
48 square units.
If $det A = - 5$ , state the determinant and area factor of $A^{- 1}$ .
2 marks
The inverse determinant is the reciprocal.
Take the absolute value for area.
Determinant $- \frac{1}{5}$ ; area factor $\frac{1}{5}$ .
Find the area effect of $(\begin{array}{cccccccccc} 1 & 2 \\ 2 & 4 \end{array})$ .
2 marks
The determinant is zero, so the map is singular.
Every image region has area zero.
The diagonal matrix $(\begin{array}{cccccccccc} k & 0 \\ 0 & 3 \end{array})$ sends an area-4 region to area 24. Find all real k.
2 marks
The area factor is 6, so $∣ 3 k ∣ = 6$ .
Thus the magnitude of k is 2.
$k = 2$ or $k = - 2$ .

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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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