Further Maths · Beta

Matrices and transformations

Change a matrix and watch its action on vectors, a unit square and area.

Drag P. The red point is its image under the matrix.

(1101)(21)=(31)\begin{pmatrix}1&1\\0&1\end{pmatrix}\begin{pmatrix}2\\1\end{pmatrix}=\begin{pmatrix}3\\1\end{pmatrix}
Drag the labelled rings or use their arrow keys.-6-4-2246-6-4-2246xy0P′P

Tab to a handle, then use the arrow keys. Use the sliders for precise values.

The image preserves orientation.

Determinant
det⁡M=1\det M=1
Area scale factor
∣det⁡M∣=1|\det M|=1
Inverse
M−1=11(1−101)M^{-1}=\frac{1}{1}\begin{pmatrix}1&-1\\0&1\end{pmatrix}

The blue square has area 1. The red parallelogram has area |det M|. Read matrix columns as the images of the two unit basis vectors.

Predict, test and explain

Step 1 of 3

Predict

What is the determinant of the shear matrix with rows (1, 1) and (0, 1)?

InvestigateMake the determinant zero. What happens to the unit square?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

What is the determinant of the shear matrix with rows (1, 1) and (0, 1)?

det⁡(1101)\det\begin{pmatrix}1&1\\0&1\end{pmatrix}

Hint

Each matrix column gives the image of one unit basis vector.

Reveal answer

1 × 1 − 1 × 0 = 1. This shear changes shape while preserving area.

11

Build an explanation

Hint 1

Each matrix column gives the image of one unit basis vector.

Hint 2

For a 2 × 2 matrix, the determinant is ad − bc.

Hint 3

The absolute determinant is the area scale factor.

Worked example

This example uses fixed values, separate from the diagram controls.

Apply the matrix with rows (1, 1), (0, 1) to (2, 1).

  1. The first output is 1 × 2 + 1 × 1 = 3.
  2. The second output is 0 × 2 + 1 × 1 = 1.
  3. Write the transformed column vector.

Answer: (3, 1).

Connect this idea

Watch out: Use ad − bc, not ad + bc. A zero determinant means the image collapses and the matrix has no inverse.