Module 4: Matrices

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Learning outcomes

After this module, you should be able to:

• write and identify matrices and their types;
• perform matrix operations and state their algebraic laws;
• compute the transpose, determinant, and inverse of a matrix;
• interpret matrices geometrically as linear / affine transforms.

Prerequisites

Complete Module 3 first. Review its quick-revision section if any term below feels unfamiliar.

Short study blocks

Treat each row as a separate lesson. Calculate the examples on paper rather than only reading them.

Block	Topic	Suggested time
1	Matrix notation, dimensions, and types	10-15 minutes
2	Addition, multiplication, and transpose	10-15 minutes
3	Determinants and inverses	10-15 minutes
4	Geometric transformations	10-15 minutes
5	Row operations and linear systems	10-15 minutes

Start here: the simple idea

A matrix is a rectangular array of numbers that we treat as a single object. We can add them, multiply them, and apply them to vectors to transform space — which is why matrices are the workhorse of computer graphics, ML, and solvers.

Everyday analogy

• A matrix is like a spreadsheet block: rows × columns of numbers.
• Matrix multiplication is like a recipe: each output cell is a dot product of one input row and one input column.
• A matrix applied to a vector is a linear transformation: stretch, rotate, or skew an image.

Matrix notation and types

An m×n matrix A has m rows and n columns; entry a_ij is row i, column j.

A = [a_ij]_{m×n}
1 2 3       ← row
4 0 5
^  ^
cols

Types:

Type	Meaning
Square	m = n
Diagonal	a_ij = 0 for i ≠ j
Identity I_n	diagonal = 1, rest 0
Zero O	all entries 0
Symmetric	A = Aᵀ
Upper/lower triangular	zeros below/above diagonal
Row vector	1 × n; Column vector

Running example:

M = [[2, 1],
     [1, 2]]   (2×2, symmetric)
A = [[1, 2, 3],
     [0, 1, 4],
     [5, 6, 0]]  (3×3)

Matrix operations

Addition and scalar multiplication

• A + B: entry-wise (same size only).
• (kA)_ij = k·a_ij.

Matrix multiplication

C = AB, where c_ij = Σ_k a_ik · b_kj (row i of A · column j of B). Requires: columns(A) = rows(B). Result size: rows(A) × columns(B).

Worked example: M · [[1],[0]]:

[[2,1],[1,2]] · [1;0] = [2·1+1·0; 1·1+2·0] = [2;1]

Transpose

(Aᵀ)_ij = a_ji; (Aᵀ)ᵀ = A; (AB)ᵀ = BᵀAᵀ; (A+B)ᵀ = Aᵀ+Bᵀ.

Matrix algebra (laws)

Law	Holds?
Commutative addition: A+B = B+A	✔
Associativity: (AB)C = A(BC)	✔
Distributivity: A(B+C) = AB+AC	✔
Commutative multiplication: AB = BA	✘ (in general)
Zero product: AB = 0 ⇔ A = 0 or B = 0	✘ (can have AB = 0 both nonzero)
Cancellation: AB = AC ⇒ B = C	✘ (without A invertible)

Determinant

For 2×2: |[[a,b],[c,d]]| = ad − bc. For 3×3: cofactor expansion along a row/column.

|a b c|           a(ei−fh) − b(di−fg) + c(dh−eg)
|d e f|,   det =
|g h i|

Key facts: det(AB) = det(A)det(B); det(Aᵀ) = det(A); A invertible ⇔ det(A) ≠ 0.

Inverse of a matrix

A⁻¹ satisfies AA⁻¹ = A⁻¹A = I. Exists iff A is square and det(A) ≠ 0.

2×2 formula

[[a,b],[c,d]]⁻¹ = 1/(ad−bc) · [[d,−b],[−c,a]]

Worked example: M = [[2,1],[1,2]], det = 4−1 = 3.

M⁻¹ = (1/3)·[[2,−1],[−1,2]]
Check: M·M⁻¹ = (1/3)·[[2·2+1·(−1), 2·(−1)+1·2],[1·2+2·(−1), 1·(−1)+2·2]]
            = (1/3)·[[3,0],[0,3]] = I  ✔

3×3 (adjoint / cofactor method)

A⁻¹ = (1/det(A)) · adj(A), where adj(A) = (cofactor matrix of A)ᵀ. On exams, Gauss-Jordan ([A | I] → [I | A⁻¹]) is usually faster.

Transpose (properties)

Already listed; note (AB)ᵀ = BᵀAᵀ (order reverses).

Geometric significance

A matrix represents a linear transformation. Interpreting M = [[2,1],[1,2]]:

• Its columns are the images of the basis vectors: e₁ → (2,1), e₂ → (1,2).
• The determinant = area scale factor. det(M) = 3, so unit area triples.
• Eigendirections (Module 6): directions that are only stretched, not rotated.
• An affine transform = x ↦ Mx + b (matrix + translation); translation is added to turn the origin, so affine needs homogeneous coordinates (3×3 for 2D).

Common 2D transforms (3×3 homogeneous form)

Rotation by θ:   [[cosθ, -sinθ, 0],[sinθ, cosθ, 0],[0,0,1]]
Scaling by s,t:   [[s,0,0],[0,t,0],[0,0,1]]
Translation (tx,ty): [[1,0,tx],[0,1,ty],[0,0,1]]

Elementary row operations and solving systems

To solve Ax = b, augment [A | b] and reduce to row echelon form (Gaussian elimination). The three elementary row operations: swap rows, scale a row, add a multiple of one row to another. These correspond to left-multiplication by elementary matrices, and A⁻¹ can be found via [A | I] → [I | A⁻¹].

Common mistakes

• Assuming AB = BA (matrix multiplication is not commutative).
• Writing (AB)ᵀ = AᵀBᵀ (wrong order — it is BᵀAᵀ).
• Thinking det(A+B) = det(A)+det(B) (determinant is not additive).
• Inverting a matrix with det = 0 (it is singular; no inverse).

Memory rules

• Entry a_ij = row i, column j; multiplication requires inner dimensions match.
• Transpose flips rows/cols: (AB)ᵀ = BᵀAᵀ (order flips).
• det(AB) = det(A)det(B); A invertible ⇔ det(A) ≠ 0.
• 2×2 inverse = 1/det · swap diagonal, negate off-diagonal.
• Matrix = linear transform; det = scale factor.

Mini Quiz

Attempt all questions before revealing the answers.

1. For which sizes is A + B defined? When is AB defined?
2. Compute [[1,2],[3,4]] · [[0,1],[−1,0]].
3. What is the transpose of [[1,2,3],[4,5,6]]?
4. When does a square matrix have no inverse?
5. What does |det(A)| = 2 mean geometrically for a 2×2 A?

Answers

Reveal answers

1. A+B: same dimensions. AB: columns of A = rows of B.
2. [[1·0+2·(−1), 1·1+2·0],[3·0+4·(−1), 3·1+4·0]] = [[−2,1],[−4,3]].
3. [[1,4],[2,5],[3,6]] (3×2).
4. When det(A) = 0 (singular).
5. Areas are scaled by a factor of 2.

Quick revision box

• Matrix = m×n array; entry a_ij (row i, col j).
• Operations: +, scalar·, multiply (row·col), transpose ((AB)ᵀ=BᵀAᵀ).
• Laws: + and × associative/distribute; × not commutative.
• det(AB)=detA·detB; A⁻¹ exists iff detA≠0; 2×2 inverse = 1/det·swap diag, flip off-diag.
• Matrix = linear transform; det = area/volume scale factor; columns = images of basis vectors.
• Gauss-Jordan: [A|I]→[I|A⁻¹].

Exam guidance

Define the notation, state the rule or theorem, show every calculation, and verify or interpret the final result.

Practice ladder

1. Easy - Recall: Define the module's central idea in one or two sentences.
2. Easy - Recognize: Identify the correct method for a small example and explain why it fits.
3. Medium - Apply: Work through one representative problem without copying the example.
4. Medium - Compare: Contrast two methods or concepts from the module.
5. Hard - Integrate: Solve a university-style scenario and justify every major step.

Reveal self-evaluation guide

A complete response uses correct terminology, shows intermediate steps, connects the result to the scenario, and states one assumption or limitation.

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