Simulic
English

2 × 2 matrices as transformations – determinant, composition, inverse and eigenvectors

MathematicsVectors & Coordinate GeometryAges 16–17

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Edit a 2 × 2 matrix, or drag the images of the two unit vectors, and watch the unit square and a shape (letter F, triangle, circle) transform, with the determinant as the signed area scale factor. Presets cover rotation, reflection, enlargement, shear, stretch and projection, plus an optional translation. The Composition screen shows that order matters (BA ≠ AB), the Inverse & systems screen solves AX = B with A⁻¹ (including 3 × 3), and the Eigenvectors screen shows eigenvectors staying on their own line and Mⁿ through diagonalisation.

Lesson: Matrices as linear transformations: matrix multiplication, determinant and area, composition, inverse matrices, solving AX = B, eigenvalues, eigenvectors and diagonalisation

What it shows

A 2 × 2 matrix M maps every point x of the plane to Mx. Its columns are the images of (1, 0) and (0, 1), so rotations, reflections, enlargements, shears and stretches each have their own matrix. The determinant ad − bc is the signed area scale factor: a negative value means the image is mirrored, and zero means the plane is squashed onto a line, so no inverse exists. Doing A and then B is the product BA. An eigenvector keeps its direction, Mv = λv, and with two distinct eigenvalues M = PDP⁻¹, which makes powers Mⁿ easy.

How to use

On Transform, pick a Matrix preset or type the four entries, drag the arrow tips î′ and ĵ′ or the point P, then press Animate I → M. On Composition, choose First: A and Then: B and compare the two orders. On Inverse & systems, pick a System and edit the coefficients. On Eigenvectors, drag v around the circle or press Sweep v, and change Power n.

Parameters you can change

  • Screen Transform, Composition, Inverse & systems, Eigenvectors
  • Matrix preset Custom, Identity matrix I, Rotation about O, Reflection in a line through O, Enlargement, centre O, Shear parallel to the x-axis, Shear parallel to the y-axis, Stretch parallel to the x-axis, Stretch parallel to the y-axis, Projection onto a line
  • Entry a (row 1, column 1) -4–4
  • Entry b (row 1, column 2) -4–4
  • Entry c (row 2, column 1) -4–4
  • Entry d (row 2, column 2) -4–4
  • Angle θ (rotation, reflection, projection) -180–180 °
  • Factor k (enlargement, shear, stretch) -3–3
  • Shape Unit square only, Letter F, Triangle, Unit circle
  • Show the transformed grid
  • Add a translation t
  • x-component of t -4–4
  • y-component of t -4–4
  • First transformation A Identity, Rotation 90°, Rotation 180°, Rotation −90°, Rotation 45°, Reflection in the x-axis, Reflection in the y-axis, Reflection in y = x, Reflection in y = −x, Enlargement ×2, Enlargement ×½, Shear along x, k = 1, Shear along y, k = 1, Stretch along x, factor 2, Stretch along y, factor 2
  • Second transformation B Identity, Rotation 90°, Rotation 180°, Rotation −90°, Rotation 45°, Reflection in the x-axis, Reflection in the y-axis, Reflection in y = x, Reflection in y = −x, Enlargement ×2, Enlargement ×½, Shear along x, k = 1, Shear along y, k = 1, Stretch along x, factor 2, Stretch along y, factor 2
  • Order shown (Composition screen) Both orders, A then B (BA), B then A (AB)
  • System of equations Unique solution (2 unknowns), Parallel lines (2 unknowns), Same line (2 unknowns), Three unknowns (3 × 3)
  • Picture of a 2 × 2 system Rows: two lines, Columns: combining vectors
  • Matrix (Eigenvectors screen) Symmetric matrix, Triangular matrix, Transition matrix (columns add to 1), Reflection in y = x, Shear (one eigenvector line), Rotation 90° (no real eigenvectors), Matrix M from the Transform screen
  • Power n of Mⁿ 1–10
  • Show the points x₀, Mx₀, M²x₀, …

Questions to explore

  1. Which matrix triples the area of the letter F without mirroring it, and which one mirrors it but keeps its area?
  2. Does a 90° rotation followed by a reflection in the x-axis give the same image as the other order, and which single transformation is each?
  3. Why does multiplying a vector by a transition matrix again and again turn it towards one eigenvector?

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