A scaling matrix has the factors on the diagonal and zeros elsewhere. Each axis stretches independently.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Rotation
A rotation by θ has columns ⟨cos θ, sin θ⟩ and ⟨−sin θ, cos θ⟩. Both are unit vectors and they stay perpendicular.
Reflection
Reflecting across the x-axis has columns ⟨1, 0⟩ and ⟨0, −1⟩. Its determinant is negative, because orientation is reversed.
Shear
A shear has rows [1, k] and [0, 1]. It slides each row of the plane sideways in proportion to its height, and preserves area.
What the determinant tells you
A determinant of 1 preserves area exactly. A negative one means the plane has been flipped over.
Building complicated ones
Any transformation can be assembled by multiplying these together. Order matters, so choose it deliberately.
Rotation
A rotation by θ has columns ⟨cos θ, sin θ⟩ and ⟨−sin θ, cos θ⟩ — the images of the two basis vectors. Deriving it from where the basis lands means it never has to be memorised.
Reflection
Reflecting across an axis negates one coordinate. Reflecting across a general line is a rotation, a reflection, and a rotation back, which is composition doing real work.
Shear
A shear slides each row sideways in proportion to its height, turning a square into a parallelogram. It preserves area, which the determinant confirms by coming out as 1.
Scaling
A diagonal matrix stretches along the axes independently. Unequal entries distort shapes, and a zero entry collapses a dimension entirely — which is precisely when the transformation stops being invertible.
Step 2: Try It Yourself
Tap and try it out.
- i-hat lands on(1, 0)
- j-hat lands on(0, -1)
- Determinant-1
- v lands on(2, -2)
The shaded parallelogram is the image of the unit square, and its area is 1. That is exactly what the determinant measures.
Step 3: Watch an Example
One step at a time.
Watch Jonas Identify a Matrix
Jonas is given the matrix with rows [1, 0] and [0, −1] and must say what it does.
- Step 1
He checks where ⟨1, 0⟩ lands, which is ⟨1, 0⟩, so the x-axis is untouched.
Step 4: Your Turn
Practice makes it stick.
The Double
Problem 1 of 2
A scaling with rows [3, 0] and [0, 3]. What is its determinant?
The Shear
Problem 2 of 2
A shear with rows [1, 4] and [0, 1]. What is its determinant?
Name That Transformation
1 of 8
Rows [2, 0] and [0, 5]. What is the determinant?
2 of 8
A reflection. Is its determinant negative? 1 yes, 0 no.
3 of 8
A rotation. What is its determinant?
4 of 8
Rows [1, 7] and [0, 1]. What is the determinant?
5 of 8
A rotation by 0 degrees. What is the entry in row 1, column 1?
6 of 8
A scaling by 4 in both directions. By what factor does area grow?
7 of 8
Match each matrix to the transformation it performs.
Tap a card on the left to start.
8 of 8
Rows [0, −1] and [1, 0]. What is the determinant?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A scaling with rows [6, 0] and [0, 2]. What is its determinant?
Question 2 of 2
What does a negative determinant mean?
What You Learned
- Scaling puts factors on the diagonal; rotation uses cosines and sines; shear slides rows sideways.
- A reflection has a negative determinant, because it turns the plane over.
- Any transformation can be built by multiplying these standard ones together.