MATH1061 · Mathematics 1a
Matrices
A matrix is a rectangular grid of numbers that adds, multiplies and — most importantly — transforms space; it is the engine behind every system, inverse and eigen-problem in the course. The central algorithm is Gaussian elimination: use elementary row operations to reduce A (or the augmented [A|b]) to row-echelon / reduced row-echelon form, then back-substitute to solve Ax = b. From there the chapter builds the inverse (when det A ≠ 0), the determinant by cofactor expansion, and the climax of the linear-algebra stream — eigenvalues and eigenvectors: solve det(A − λI) = 0 for the eigenvalues, find the eigenspaces, and diagonalise A = PDP⁻¹, whose payoff is fast matrix powers. Everything is done by hand with exact arithmetic.
What this chapter covers
- 01B9 Matrix operations — addition, scalar multiple, the dimension rule for products
- 02B8 Gaussian elimination — the three row operations → REF / RREF
- 03B8 Solving Ax = b by row-reduction and back-substitution
- 04B10 Transpose and the inverse (2×2 and row-reduction methods)
- 05B11 The determinant by cofactor expansion
- 06B12–B13 Eigenvalues, eigenvectors and diagonalisation A = PDP⁻¹
Worked example: eigenvalues and eigenvectors of a 2×2 matrix
- +1Form A − λI: [[2 − λ, 1], [1, 2 − λ]].
- +1Characteristic equation: det(A − λI) = (2 − λ)² − 1 = 0.
- +1Solve: (2 − λ)² = 1 ⇒ 2 − λ = ±1 ⇒ λ = 1 or λ = 3.
- +1Eigenvector for λ = 3: (A − 3I)v = 0 gives [[−1, 1],[1, −1]]v = 0 ⇒ v₁ = v₂, so v = (1, 1).
- +1Eigenvector for λ = 1: (A − I)v = 0 gives [[1, 1],[1, 1]]v = 0 ⇒ v₁ = −v₂, so v = (1, −1).
- +1Check: A(1, 1) = (3, 3) = 3(1, 1) ✓ and A(1, −1) = (1, −1) = 1·(1, −1) ✓.
Key terms
- Gaussian elimination
- The algorithm that uses the three elementary row operations (swap rows, scale a row, add a multiple of one row to another) to reduce a matrix to row-echelon form, then reduced row-echelon form — the core routine for solving systems, inverting, and finding rank.
- Reduced row-echelon form (RREF)
- The unique fully-reduced form: leading 1 in each non-zero row, each leading 1 to the right of the one above, and zeros above and below each leading 1. The solution of Ax = b can be read straight off the RREF of [A|b].
- Inverse
- The matrix A⁻¹ with A A⁻¹ = I, which exists exactly when det A ≠ 0. For 2×2, A⁻¹ = (1/det A)[[d, −b], [−c, a]]; in general, row-reduce [A | I] to [I | A⁻¹].
- Determinant
- A single scalar det A computed by cofactor expansion (or the ad − bc rule for 2×2). It is zero exactly when A is singular (non-invertible) and equals the scaling factor of area/volume under the transformation A.
- Eigenvalue / eigenvector
- A scalar λ and non-zero vector v with Av = λv — a direction the matrix only stretches, never rotates. Eigenvalues solve det(A − λI) = 0; the eigenvectors span the eigenspaces and, when there are enough, diagonalise A = PDP⁻¹.
Matrices FAQ
How do I solve a system Ax = b?
Form the augmented matrix [A | b], then use Gaussian elimination to reduce it to row-echelon (or reduced row-echelon) form. Back-substitute to read off the solution. The shape of the RREF tells you everything: a unique solution, infinitely many (a free variable / row of zeros with consistent right side), or no solution (a row 0 = nonzero).
When does a matrix have an inverse?
Exactly when its determinant is non-zero. A zero determinant means the matrix is singular — it collapses dimensions, so the transformation can't be undone and Ax = b either has no solution or infinitely many. For a 2×2 matrix the inverse is (1/det)[[d, −b], [−c, a]]; for larger matrices, row-reduce [A | I] to [I | A⁻¹].
How do I find eigenvalues and eigenvectors?
Solve the characteristic equation det(A − λI) = 0 for the eigenvalues λ. For each λ, substitute back and solve the homogeneous system (A − λI)v = 0 to find the eigenvectors (the eigenspace). Always check Av = λv at the end — it catches arithmetic slips, and it's a guaranteed method mark.
What is diagonalisation good for?
If A has enough independent eigenvectors, you can write A = PDP⁻¹ where D is the diagonal matrix of eigenvalues and P's columns are the eigenvectors. The payoff is fast powers: Aⁿ = PDⁿP⁻¹, and Dⁿ is just each eigenvalue raised to n. It also decouples the action of A into independent stretches along the eigenvector directions.
Exam move
Drill Gaussian elimination until it is automatic — it is the spine of the whole chapter and reused for solving, inverting and finding rank. Keep the row operations clean and labelled so a single slip doesn't cascade. For the inverse, check det ≠ 0 first; for 2×2 use the ad − bc formula, otherwise row-reduce [A | I]. For the eigen-problem, follow the fixed chain: det(A − λI) = 0 → eigenvalues → (A − λI)v = 0 → eigenvectors → (if asked) diagonalise, and always verify Av = λv. Because the work is by hand and partly no-calculator, keep entries exact and show every row operation — the method marks reward the process.
