SME
Grade 12 Math · Problem type
How a matrix changes area
Use this for “A shape of area 4 is transformed by [[3, 1], [1, 2]]. What is the new area?”
Chart
Determinant of A: 5
Area before: 4
Area after: 20
This is an example. Type your own values to start a new problem.
=
det A
T
=
|
det A
|
×
S
Formulas
det
A
=
a
×
d
−
b
×
c
5 = 3 × 2 − 1 × 1
T
=
|det
A|
×
S
20 = |5| × 4
Assumptions
•
The unit square lands on the parallelogram with sides (a, c) and (b, d), the matrix’s columns.
•
Its area is |det A| = |ad − bc|, and every area is multiplied by the same |det A|.
•
A negative det A also flips the figure over; det A = 0 squashes it flat.
Step-by-step
WE KNOW
a = 3, b = 1, c = 1, d = 2, S = 4
FIND
determinant of A (det A), area after (T)
Step 1 · Find determinant of A (det A)
Use det A = a × d − b × c
The area of the unit square’s image, with a sign for flipping.
det A = 3 × 2 − 1 × 1
det A = 6 − 1
det A = 5
Step 2 · Find area after (T)
Use T = |det A| × S
Every unit square becomes a parallelogram of area |det A|, so every area is multiplied by |det A|.
T
=
|5|
×
4
T
=
5
×
4
T
=
20
Check
Put the numbers back in. Both sides should match.
5 = 3 × 2 − 1 × 1 ✓
20 = |5| × 4 ✓
Topic check
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Related lessons
Matrices as transformations of the plane
Main lesson
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