A matrix is just a rectangular grid of numbers, and it turns out to be one of the most useful ideas in maths: the same grid can store a list of coordinates, describe a reflection, or run a whole 3D animation frame in a video game. In geometry, a matrix is a compact machine for moving points around a plane — and once you can multiply matrices, reflecting, rotating and enlarging a shape (which you already met in g8 Transformations) becomes a single calculation instead of a drawing exercise.
1. What a matrix is: order, rows and columns
A matrix is written inside big brackets. Its order is given as rows × columns — always rows first. The matrix A = [[2, 1], [3, 4]] has 2 rows and 2 columns, so it has order 2×2. A single column of numbers like [[3], [5]] is a column vector, order 2×1 — exactly the kind of object used to store a point's (x, y) coordinates.
| Matrix | Rows | Columns | Order |
|---|---|---|---|
| [[2, 1], [3, 4]] | 2 | 2 | 2×2 |
| [[5], [ -2]] | 2 | 1 | 2×1 (column vector) |
| [[1, 0, 3]] | 1 | 3 | 1×3 (row vector) |
2. Adding, subtracting and scalar multiplying
- You can only add or subtract two matrices of the same order — add each pair of matching entries. [[2, 1], [3, 4]] + [[1, 0], [2, 3]] = [[3, 1], [5, 7]].
- Scalar multiplication means multiplying every entry by a single number. 3 × [[2, 1], [3, 4]] = [[6, 3], [9, 12]].
- There is no such thing as adding a 2×2 matrix to a 2×1 matrix — the orders must match exactly.
3. Multiplying matrices: row × column
Matrix multiplication is not "multiply matching entries" — it is the row-times-column method. To find the entry in row i, column j of the answer, take row i of the first matrix, take column j of the second matrix, multiply the pairs together and add.
Top-left (row 1 of A · column 1 of B): (2×1) + (1×2) = 2 + 2 = 4.
Top-right (row 1 of A · column 2 of B): (2×0) + (1×3) = 0 + 3 = 3.
Bottom-left (row 2 of A · column 1 of B): (3×1) + (4×2) = 3 + 8 = 11.
Bottom-right (row 2 of A · column 2 of B): (3×0) + (4×3) = 0 + 12 = 12.
So AB = [[4, 3], [11, 12]].
The same method multiplies a 2×2 matrix by a 2×1 column vector — this is exactly how a transformation is applied to a point. [[1, 0], [0, -1]] × [[3], [2]]: top = (1×3)+(0×2) = 3, bottom = (0×3)+(-1×2) = -2, giving the point (3, -2).
4. The determinant and the inverse of a 2×2 matrix
For M = [[a, b], [c, d]], the determinant is det(M) = ad − bc, a single number that measures how much the matrix scales area (and whether it flips orientation). If det(M) = 0, the matrix has no inverse — it is called singular, because it squashes every point onto a line, so the transformation cannot be undone.
When det(M) ≠ 0, the inverse M-1 undoes M's transformation. The rule: swap the entries on the leading diagonal (a and d), negate the entries on the other diagonal (b and c), then divide every entry by the determinant:
M-1 = (1/det M) × [[d, -b], [-c, a]]
Put these steps in the correct order for finding the inverse of a 2×2 matrix:
- Calculate the determinant, ad − bc
- Check the determinant is not zero (otherwise there is no inverse)
- Swap a and d, and negate b and c
- Multiply every entry of the new matrix by 1/det
det(M) = (3×4) − (2×1) = 12 − 2 = 10.
Swap the diagonal (3 and 4) and negate the other diagonal (2 and 1): [[4, -2], [-1, 3]].
M-1 = (1/10) × [[4, -2], [-1, 3]] = [[0.4, -0.2], [-0.1, 0.3]].
5. Matrices as transformations — the g8 toolkit, rebuilt
In g8 Transformations you reflected, rotated and enlarged shapes by counting squares on a grid. Every one of those moves can instead be written as a 2×2 matrix that you multiply by each vertex's column vector (x, y):
| Transformation | Matrix |
|---|---|
| Reflection in the x-axis | [[1, 0], [0, -1]] |
| Reflection in the y-axis | [[-1, 0], [0, 1]] |
| Reflection in the line y = x | [[0, 1], [1, 0]] |
| Rotation 90° anticlockwise about the origin | [[0, -1], [1, 0]] |
| Rotation 180° about the origin | [[-1, 0], [0, -1]] |
| Enlargement, scale factor k, centre origin | [[k, 0], [0, k]] |
Multiply the transformation matrix by a point's column vector to move it: reflecting (3, 2) in the x-axis with [[1, 0], [0, -1]] gives [[1×3+0×2], [0×3+(-1)×2]] = (3, -2) — exactly the mirror image you'd get counting squares in g8, just reached by calculation instead of a diagram. Notice the enlargement matrix's determinant is k²: an enlargement scale factor k scales area by k², which is exactly what the determinant measures.
Exam technique: write each matrix's order in small brackets next to it before you multiply, e.g. A(2×2) × B(2×1). If the inner numbers (2 and 2) don't match, the multiplication is impossible — and if they do match, the outer numbers tell you the order of the answer (here, 2×1) before you've calculated a single entry.