Matrix Algebra and Transformations
Matrices
A matrix is a rectangular array of numbers, described by its order (rows × columns). Matrices represent systems of equations and geometric transformations.
Matrix arithmetic
- Add/subtract: only for matrices of the same order; add corresponding elements.
- Scalar multiplication: multiply every element by the scalar.
- Matrix multiplication: multiply row by column. For AB, the number of columns of A must equal the number of rows of B. Multiplication is not commutative (AB ≠ BA in general).
Example (2×2):
[a b][e f] [ae+bg af+bh]
[c d][g h] = [ce+dg cf+dh]The identity matrix
The identity matrix I has 1s on the leading diagonal and 0s elsewhere. It behaves like "1": AI = IA = A.
Matrices as transformations
A 2×2 matrix transforms points (vectors) in the plane. Multiply the matrix by the position vector to get the image. Standard transformations:
| Transformation | Matrix |
|---|---|
| Rotation by θ (anticlockwise, about origin) | [cos θ −sin θ; sin θ cos θ] |
| Reflection in x-axis | [1 0; 0 −1] |
| Reflection in y-axis | [−1 0; 0 1] |
| Reflection in y = x | [0 1; 1 0] |
| Enlargement scale factor k | [k 0; 0 k] |
Combining transformations
Applying transformation P then Q is represented by the product QP (the second transformation is written first, because it multiplies the result of the first). Order matters.
Invariant points and lines
An invariant point maps to itself under a transformation (e.g. the origin for a rotation about the origin). An invariant line maps onto itself.
Worked example
Find the image of the point (2, 3) under the matrix [1 0; 0 −1] (reflection in the x-axis).
- [1 0; 0 −1][2; 3] = [1×2 + 0×3; 0×2 + (−1)×3] = (2, −3) — the y-coordinate is negated, as expected for a reflection in the x-axis. ✓
Common mistakes
- Multiplying matrices in the wrong order (AB ≠ BA).
- Combining transformations as PQ instead of QP (do first transformation last in the product).
- Trying to multiply matrices with incompatible orders.
Exam tips
- Check the orders before multiplying (columns of A = rows of B).
- Learn the standard transformation matrices, especially rotation.
- For combined transformations, write the second transformation matrix first.
Key facts to remember
- Matrix multiplication: row × column, columns of A = rows of B, not commutative; I is the identity (AI = A).
- 2×2 matrices represent transformations; learn rotation [cos θ −sin θ; sin θ cos θ], reflections, enlargements.
- Combined transformation P then Q = QP; watch the order and find invariant points/lines.