A 2×2 matrix as a machine that moves the whole plane

How to use this simulation

1. The matrix [[a, b], [c, d]] has four sliders, one per entry. At the start it is [[2, 1], [0.5, 1.5]]. The pale grid is the original plane; the blue grid is the same plane after the matrix has acted on it. The play button repeats the glide from the identity to the matrix.
2. The yellow square is the unit square, with corners (0, 0), (1, 0), (1, 1), (0, 1). The small purple “F” sits inside it. An F is useful because its mirror image is easy to spot.
3. The green arrow is î (the arrow (1, 0)) after the transformation, and the purple arrow is ĵ (the arrow (0, 1)). They land at the columns of the matrix: here î goes to (2, 0.5) and ĵ to (1, 1.5).
4. Pick a ready-made transformation: rotation (use the angle slider), reflections, shear, scale, or a squash with determinant 0. Moving any entry slider switches to your own custom matrix.
5. The number written on the square is the determinant, the signed area of the transformed unit square. Here it is 2.5. When it goes negative the square turns red and the F becomes a mirror image.
6. The readings give the inverse matrix whenever one exists. If the determinant is 0 the square is flattened onto a line, and the inverse is “none”.

A matrix is a machine for points

Take a point (x, y). A 2×2 matrix [[a, b], [c, d]] turns it into a new point (ax + by, cx + dy). That is all multiplication by a matrix means: a recipe that mixes x and y in a fixed way.

Now do it to every point at once. The grid lines stay straight, parallel lines stay parallel, and the origin stays put. These are the “linear” transformations. Anything fancier, like bending a line into a curve, is not a matrix job.

The two columns tell you everything

Every point (x, y) is x steps along î and y steps along ĵ. A linear transformation keeps that recipe: it only moves î and ĵ, and everything else follows. So to know the whole transformation, just ask where î = (1, 0) and ĵ = (0, 1) land.

Put (1, 0) into the matrix and you get (a, c), the first column. Put in (0, 1) and you get (b, d), the second column. In the simulation the starting matrix [[2, 1], [0.5, 1.5]] sends î to (2, 0.5) and ĵ to (1, 1.5). Those are exactly the green and purple arrows.

This is the best way to build a matrix from a picture. Want a transformation that sends î up to (0, 1) and ĵ to (−1, 0)? Write those two arrows as columns: [[0, −1], [1, 0]]. That one is a 90° turn.

The standard transformations

A handful of matrices do almost all the work in exams and in graphics. The table lists seven of them with their determinants. Rotation by 90° anticlockwise comes first.

• Rotation by θ anticlockwise: [[cos θ, −sin θ], [sin θ, cos θ]]. Lengths and angles do not change.
• Reflection in the x axis flips the sign of y. Reflection in the y axis flips x. Reflection in the line y = x swaps x and y.
• Shear [[1, k], [0, 1]] slides each row sideways in proportion to its height. The x axis stays; the y axis leans.
• Scale [[p, 0], [0, q]] stretches by p along x and by q along y.
• The last row is a squash: the columns point the same way, so the whole plane lands on one line.

The determinant: signed area

The unit square has area 1. After the transformation it becomes a parallelogram with sides (2, 0.5) and (1, 1.5). Its area is the determinant ad − bc. For the starting matrix that is 2.5, so every shape on the plane grows to 2.5 times its old area.

The determinant works for every shape, not just the unit square. A matrix with determinant 3 triples every area, a circle’s or a cat’s. A determinant of 0.5 halves it.

The sign carries extra news. If the determinant is negative, the transformation has flipped the plane over like a sheet of paper. A clockwise F becomes an anticlockwise one. That is the mirror image you see in the simulation, and the reason the square turns red.

• Positive: orientation kept. Bigger than 1 stretches areas, between 0 and 1 shrinks them.
• Negative: orientation flipped, and areas scale by the size without the sign.
• Zero: everything is squashed onto a line (or a point). Area is lost, and nothing can undo it.

The inverse: running the machine backwards

The inverse matrix undoes the transformation. If A takes a point to a new place, A⁻¹ takes it back. For a 2×2 matrix the recipe is short: swap a and d, flip the signs of b and c, then divide everything by the determinant.

That division is why a determinant of 0 kills the inverse. Take the starting matrix [[2, 1], [0.5, 1.5]] with determinant 2.5: its inverse is [[0.6, -0.4], [-0.2, 0.8]]. You can read this off the readings panel.

Think about it as pictures too. If the transformation squashes the whole plane onto a line, many different points land on the same spot. Going backwards would need to choose among them, so there is no way to undo it.

Doing two transformations: order matters

To do transformation Q first and then P, multiply the matrices as P·Q, with the first one to act written on the right. That feels backwards, but it matches how we write functions: P(Q(x)).

Matrix multiplication is not commutative. Rotating and then reflecting does not give the same result as reflecting and then rotating. Problem 5 below shows it with real numbers.

The determinant behaves nicely, though: det(PQ) = det(P) × det(Q). Two area scalings simply multiply.

Try it in the simulation

Six quick experiments, a minute each.

• Press “Identity”. Nothing moves, the determinant is 1.
• Press “Rotation” and slide the angle. The grid turns rigidly and the determinant stays at 1 the whole time.
• Press “Reflect in x axis”. The F flips upside down and the square goes red: determinant −1.
• Press “Shear”. The unit square leans into a parallelogram with the same area, determinant 1.
• Press “Scale” and compare the determinant with the product of the two stretch factors.
• Press “Squash”. The determinant is 0, the grid collapses onto a line, and the readings say there is no inverse.

Solved problems

Six common questions. Every number below is computed with the same functions as the simulation.

Mistakes to avoid

These cost marks again and again.

• Reading the columns as rows. The first column is where î goes, not the first row.
• Multiplying in the wrong order. PQ means Q first, then P, and PQ is usually not QP.
• Forgetting to divide by the determinant when finding the inverse.
• Writing ad + bc instead of ad − bc.
• Saying a negative determinant means “negative area”. Area is positive; the sign records the flip.
• Looking for the inverse of a matrix whose determinant is 0.

Where matrices are used

Every game and animation moves its characters with matrices: turn, flip, stretch, repeat. Photo apps use shear and scale matrices to fix a tilted picture. The same maths, in more dimensions, runs 3D graphics and robot arms.

In data science a matrix transformation is how a model mixes its inputs, and determinants show up in solving simultaneous equations, in the area of a region and in how a change of variable rescales an integral.

Exam corner

For HSC higher maths and admission tests, remember these.

• Image of a point: multiply the matrix by the column (x, y).
• Columns of the matrix are the images of î and ĵ. Use this to write a matrix from a description.
• det = ad − bc. Singular means det = 0, with no inverse.
• Inverse of a 2×2: swap a and d, negate b and c, divide by det.
• Area scale factor is |det|; a negative det flips orientation.

Frequently asked questions