2 × 2 Matrices as
Plane Transformations
Learn how a 2 × 2 matrix can rotate, reflect, scale, and shear every point in the plane, and discover how the determinant tells you exactly how area changes.
What Is a Plane Transformation?
A transformation of the plane is a function that takes every point \((x, y)\) and maps it to a new point \((x', y')\). When this function can be described by multiplying a 2 × 2 matrix by the coordinate vector, we call it a linear transformation.
The Matrix-Vector Product
To transform a point \((x, y)\), compute:
Each output coordinate is a linear combination of \(x\) and \(y\). The first row gives \(x'\), the second row gives \(y'\).
Columns Are Transformed Basis Vectors
Common Transformation Types
Rotation vs Reflection
| Rotation by \(\theta\) | Reflection over \(x\)-axis | |
|---|---|---|
| Matrix | \(\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\) | \(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\) |
| Determinant | \(+1\) | \(-1\) |
| Orientation | Preserved | Reversed |
| Area change | None (\(|\det| = 1\)) | None (\(|\det| = 1\)) |
The Determinant and Area
For a matrix \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\), the determinant is:
The absolute value \(|\det(A)|\) tells you the area scale factor: if a region has area \(S\), its image has area \(|\det(A)| \cdot S\).
Visualizing the Area Factor
How to Apply a Matrix Transformation
Example 1: Scaling
Apply \(A = \begin{bmatrix} 3 & 0 \\ 0 & 2 \end{bmatrix}\) to the point \((1, 4)\).
The point moves from \((1, 4)\) to \((3, 8)\): stretched by factor 3 horizontally and 2 vertically.
Example 2: 90° Rotation
Rotate the point \((5, 2)\) counterclockwise by \(90°\).
\(|\det| = |0 \cdot 0 - (-1)(1)| = 1\): rotation preserves area.
Example 3: Shear and Area
A triangle with vertices \((0,0)\), \((4,0)\), \((0,3)\) has area \(\frac{1}{2}(4)(3)=6\). Apply the shear \(A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}\).
When \(\det(A) = \) 0
Composing Two Transformations
Applying matrix \(B\) then matrix \(A\) is the same as applying the product \(AB\) in one step:
The area factor of the composition is:
So area factors multiply. If \(A\) doubles area and \(B\) triples it, then \(AB\) scales area by \(6\).
Common Mistakes
Summary of Transformations
| Transformation | Matrix | \(|\det|\) | Orientation |
|---|---|---|---|
| Rotation by \(\theta\) | \(\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\) | 1 | Preserved |
| Scaling by \(k_x, k_y\) | \(\begin{bmatrix} k_x & 0 \\ 0 & k_y \end{bmatrix}\) | \(|k_x k_y|\) | Depends on sign |
| Reflection (\(x\)-axis) | \(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\) | 1 | Reversed |
| Shear (horizontal) | \(\begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}\) | 1 | Preserved |
| Singular | Any with \(\det = 0\) | 0 | Collapsed |
Question 1 of 5
What is the image of the point \((3, -1)\) under the transformation \(A = \begin{bmatrix} 2 & 0 \\ 0 & 5 \end{bmatrix}\)?
Question 2 of 5
A triangle has area 10. If it is transformed by \(A = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix}\), what is the area of the image?
Question 3 of 5
Which matrix represents a counterclockwise rotation by \(180°\)?
Question 4 of 5
If \(\det(A) = -7\), what can you conclude about the transformation?
Question 5 of 5
Transformation \(A\) has \(|\det(A)| = 3\) and transformation \(B\) has \(|\det(B)| = 4\). If you apply \(B\) first and then \(A\), by what factor does area change?