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\(2 \times 2\) Matrices as
Plane Transformations

Learn how matrix multiplication transforms shapes in the coordinate plane, and discover what the determinant tells us about area.

Precalculus

What Is a Linear Transformation?

A linear transformation is a function that maps every point \((x, y)\) in the plane to a new point \((x', y')\) using the rule:

This means every point in the plane gets moved according to the same matrix. Lines stay as lines, and the origin stays fixed.

How the Matrix Maps Points

The matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) acts on a point \((x,y)\) via:

x y
Original
Transformed
Apply
\(\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}\)

A unit square becomes a parallelogram whose shape is determined entirely by the four entries of the matrix.

Reading the Column Vectors

The columns of the matrix tell you where the standard basis vectors land after the transformation:

\(\hat{\mathbf{i}} = (1,0)\)
Maps to the first column: \((a, c)\)
\(\hat{\mathbf{j}} = (0,1)\)
Maps to the second column: \((b, d)\)

The Determinant and Area

The absolute value of the determinant tells you the area scaling factor. If a region has area \(A\), then after the transformation its area becomes:

If \(\det(M) < 0\), the transformation also reverses orientation (flips the plane).

Visualizing Area Scaling

\(A=1\)
 Unit Square
\(M\)
\(A' = |\det(M)|\)
 Transformed
If \(M = \begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}\), then:
\(\det(M) = 2(1) - 1(0) = 2\)
Area scale factor is \(|2| = 2\)

Common Transformation Matrices

Rotation by \(\theta\)
\(\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\)
Scaling
\(\begin{bmatrix} k_x & 0 \\ 0 & k_y \end{bmatrix}\)
Reflection (y-axis)
\(\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\)

Rotation vs. Reflection

Rotation Reflection
Determinant \(+1\) \(-1\)
Orientation Preserved Reversed
Area Unchanged Unchanged
Example \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\) \(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)
Geometric effect Turns shapes Flips shapes

How to Apply a Matrix Transformation

1
Identify the matrix \(M = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) and the point or shape to transform.
2
Multiply: for each vertex \((x,y)\), compute \(x' = ax + by\) and \(y' = cx + dy\).
3
Plot the new points to see the transformed shape.
4
Compute the determinant \(ad - bc\) to find the area scaling factor.
5
Interpret: \(|\det| > 1\) means area expands, \(|\det| < 1\) means area shrinks, \(\det < 0\) means orientation flips.

Example 1: Scaling Transformation

Apply \(M = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}\) to the unit square with vertices \((0,0)\), \((1,0)\), \((1,1)\), \((0,1)\).

x y 1 2 1 2 3
\(A=1\)
\(A' = 6\)
 (0,0) (2,0) (2,3) (0,3)
\(\det = (2)(3) - (0)(0) = 6\)
\(|\det| = 6\)
Area scales by factor of 6

Example 2: Shear Transformation

Apply the shear matrix \(M = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}\) to the unit square.

x y 1 2 3 1 (1,0) (3,1) (2,1)
\(\det = (1)(1) - (2)(0) = 1\)
\(|\det| = 1\)
Area is preserved!

Example 3: Singular Matrix

Apply \(M = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}\) to the unit square. What happens?

1 2 3 1 2 (0,0) (1,2)
\(\det = (1)(4) - (2)(2) = 0\)
Singular matrix!
All points collapse
onto the line \(y = 2x\)
\(\text{Area} = 0\)

Composing Two Transformations

Applying transformation \(A\) then \(B\) is the same as applying the product \(BA\):

And the determinant of the composition is the product of the individual determinants:

So the total area scaling factor is \(|\det(B)| \cdot |\det(A)|\).

Common Mistakes to Avoid

✕ Wrong Determinant Formula
Computing \(ab - cd\) instead of \(ad - bc\). Remember: main diagonal minus off-diagonal.
✕ Forgetting Absolute Value
Area is always non-negative. A negative determinant means orientation flips, but area = \(|\det|\).
✕ Composition Order
Applying \(A\) then \(B\) requires multiplying \(BA\), not \(AB\). Matrix multiplication is not commutative.

Summary of Key Transformations

Transformation Matrix \(|\det|\) Effect on Area
Rotation by \(\theta\) \(\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\) 1 Preserved
Scaling by \(k\) \(\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}\) \(k^2\) Multiplied by \(k^2\)
Shear \(\begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}\) 1 Preserved
Reflection (x-axis) \(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\) 1 Preserved (flipped)
Projection onto x-axis \(\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\) 0 Collapsed to line

Question 1 of 5

What is the determinant of \(\begin{bmatrix} 4 & 2 \\ 3 & 5 \end{bmatrix}\)?

Question 2 of 5

A triangle with area 4 is transformed by a matrix with determinant -3. What is the area of the image?

Question 3 of 5

Where does \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\) send the point \((3, 2)\)?

Question 4 of 5

Which matrix represents a reflection over the x-axis?

Question 5 of 5

If \(\det(A) = 4\) and \(\det(B) = -2\), what is the area scaling factor when you apply \(A\) then \(B\)?