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Pre-Calculus Matrices

2×2 matrices as transformations of the plane

20 practice questions 0 video lessons Theory + worked examples

2×2 Matrices as Transformations

California Pre-Calculus • Standard N-VM.11 • Matrices

2×2 Matrices as Transformations is a topic in Matrices in the California Common Core State Standards. It is aligned to Standard N-VM.11, which requires students to work with 2×2 matrices as transformations of the plane and interpret the determinant as an area factor.

A \(2\times 2\) matrix transforms the plane — scaling, rotating, or reflecting points — with the determinant giving the factor by which area is scaled.

California Pre-Calculus › Matrices › 2×2 Matrices as Transformations  —  Standard N-VM.11

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Theory

A \(2\times 2\) matrix can be viewed as a transformation of the plane: multiply it by a point (written as a column vector) to get the image point.

Standard transformations have standard matrices:

  • Scaling by \(k\): \(\begin{bmatrix}k&0\\0&k\end{bmatrix}\).
  • Rotation by \(90^\circ\): \(\begin{bmatrix}0&-1\\1&0\end{bmatrix}\).
  • Reflection across the \(x\)-axis: \(\begin{bmatrix}1&0\\0&-1\end{bmatrix}\).
The determinant is the area scale factor: a unit square maps to a region of area \(|\det|\).
A 2x2 matrix transforms the plane A 2 by 2 matrix maps points to new points; here a rotation sends a horizontal vector to a vertical one. x y (2,0) image
A matrix maps each point to a new point — here a rotation.
Common transformations Common transformations Common transformations scale k: [k 0; 0 k] rotate 90°: [0 −1; 1 0] reflect over x: [1 0; 0 −1]
Standard transformation matrices.

Apply a transformation by multiplication:

\[\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}ax+by\\cx+dy\end{bmatrix}\]
multiply the matrix by the point to get the transformed point
Composition of transformations is matrix multiplication — apply the rightmost matrix first.

How to transform a point

  1. Write the point as a column vector.
  2. Multiply the transformation matrix by it.
  3. Read the image point from the result.
  4. Interpret the effect (scale, rotate, reflect) and use \(|\det|\) for the area change.
Example 1 — Apply a matrix to a point
Apply \(\begin{bmatrix}2&0\\0&2\end{bmatrix}\) to the point \((3,1)\).
Solution

Multiply the matrix by the column vector.

\[\begin{bmatrix}2&0\\0&2\end{bmatrix}\begin{bmatrix}3\\1\end{bmatrix}=\begin{bmatrix}6\\2\end{bmatrix}\]

The point scales out to \((6,2)\).

the point maps to 6 comma 2
Example 2 — A rotation
Where does \(\begin{bmatrix}0&-1\\1&0\end{bmatrix}\) send \((1,0)\)?
Solution

Multiply.

\[\begin{bmatrix}0&-1\\1&0\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix}=\begin{bmatrix}0\\1\end{bmatrix}\]

This is a \(90^\circ\) counterclockwise rotation: \((1,0)\to(0,1)\).

the point rotates 90 degrees to 0 comma 1
Example 3 — A reflection
Apply \(\begin{bmatrix}1&0\\0&-1\end{bmatrix}\) to \((4,3)\).
Solution

Multiply.

\[=\begin{bmatrix}4\\-3\end{bmatrix}\]

The \(y\)-coordinate flips — a reflection across the \(x\)-axis.

the point reflects to 4 comma negative 3
Example 4 — Determinant as area scale
By what factor does \(\begin{bmatrix}3&0\\0&2\end{bmatrix}\) scale area?
Solution

The area factor is \(|\det|\).

|\det|\(=\)\(|3\cdot 2|=6\)
area scales by a factor of 6

Common pitfalls

Point on the right. Multiply matrix times column vector, in that order.
Compose right-to-left. \(AB\) applies \(B\) first, then \(A\).
Negative determinant flips orientation (a reflection), while \(|\det|\) still gives the area scale.

Frequently asked questions

How does a 2x2 matrix transform a point?

Multiply the matrix by the point written as a column vector; the result is the image point.

What matrix rotates the plane 90 degrees?

\(\begin{bmatrix}0&-1\\1&0\end{bmatrix}\) rotates points \(90^\circ\) counterclockwise.

What does the determinant mean for a transformation?

Its absolute value is the factor by which area is scaled; a negative determinant also flips orientation.

How do you combine two transformations?

Multiply their matrices. The product applies the right-hand transformation first, then the left.

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