Matrices as transformations
A 2×2 matrix moves every point of the plane. Its columns are where i and j land, its determinant is how areas scale, and its eigenvectors are the lines that stay put.
Readouts
What's happening
A 2×2 matrix M sends each point (x, y) to M(x, y), and because the map is linear, all you need to know is where the two unit vectors go: the first column of M is the image of i and the second column is the image of j. The grid follows, squares become parallelograms, and the area of every shape is multiplied by |det M|. A negative determinant means the plane has been flipped over (look at the F), and a zero determinant squashes everything onto a line or a point, so there is no inverse. An eigenvector v is a direction the matrix does not turn: Mv = λv, so the line through the origin along v is an invariant line, stretched by the eigenvalue λ. When λ = 1 every point on it stays exactly where it is, a line of invariant points. Rotations by anything other than 0° or 180° have complex eigenvalues and no invariant lines through the origin. Doing B then A is the single matrix AB, which is usually not the same as BA.
A-Level Further Maths (AQA, Edexcel, OCR, OCR MEI): matrices as linear transformations, determinants as area scale factors, invariant points and lines, inverse and composite transformations. First-year university linear algebra: eigenvalues and eigenvectors.
Work through the numbers with Graphing Calculator and Equation Solver.
Challenge
Predict first: the shear M = (1 2 ; 0 1). What is its determinant, and which line is made of invariant points? Pick Shear, set k = 2, and check the readouts and the gold line.
det M = 1 × 1 − 2 × 0 = 1, so areas do not change even though the square leans over. The only eigenvalue is 1 with eigenvector (1, 0), so the x-axis (y = 0) is a line of invariant points. Every horizontal line is also mapped to itself, but those points slide along it.
FAQ
- How do I find the matrix of a transformation?
- Work out where the unit vectors go. The image of i = (1, 0) is the first column and the image of j = (0, 1) is the second column. A rotation by θ anticlockwise sends i to (cos θ, sin θ) and j to (−sin θ, cos θ), so its matrix is (cos θ −sin θ ; sin θ cos θ).
- What does the determinant tell me?
- |det M| is the area scale factor: every shape's area is multiplied by it. The sign tells you about orientation: positive keeps it, negative reflects it, and 0 means the plane is squashed onto a line or a point and the matrix has no inverse.
- What is the difference between an invariant line and a line of invariant points?
- On a line of invariant points every single point stays where it is (eigenvalue 1). On an invariant line the line as a whole maps onto itself, but points can move along it. The lines through the origin of both kinds lie along eigenvectors.
- Why is AB not equal to BA?
- AB means do B first and then A. Rotating then reflecting usually lands a shape somewhere different from reflecting then rotating, so the products differ. Turn on composition in Advanced and switch the order to see it.
- How do I find the matrix that maps two points to their images?
- Put the points in the columns of P and their images in the columns of Q, then M = QP⁻¹. If (1, 1) maps to (3, 1) and (1, −1) maps to (1, 3), then P = (1 1 ; 1 −1), det P = −2 and M = (2 1 ; 2 −1). It only works when the two points are not on one line through the origin. Matrix from two points under Solve for shows each step.