15.1 Matrices and Their Types
An m × n matrix is a rectangular array of numbers with m rows and n columns; A[i][j] denotes the entry in row i, column j.
| Type | Definition |
|---|---|
| Row / column matrix | 1 × n / m × 1 |
| Square | m = n |
| Diagonal | square, non-diagonal entries 0 |
| Scalar / Identity I | diagonal with equal entries / all 1s |
| Zero matrix O | all entries 0 |
| Transpose A^T | rows and columns swapped: A^T[i][j] = A[j][i] |
| Symmetric / Skew-symmetric | A^T = A / A^T = −A |
15.2 Matrix Operations
- Addition/subtraction: entrywise; only for identical dimensions.
- Scalar multiplication: multiply every entry.
- Multiplication: A (m × n) times B (n × p) gives C (m × p) with C[i][j] = Σ (k = 1..n) A[i][k]·B[k][j] — row i of A dotted with column j of B. Inner dimensions must match.
Worked check that AB ≠ BA. Let A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]].
- AB = [[1·0 + 2·1, 1·1 + 2·0], [3·0 + 4·1, 3·1 + 4·0]] = [[2, 1], [4, 3]] (B swaps A's columns).
- BA = [[0·1 + 1·3, 0·2 + 1·4], [1·1 + 0·3, 1·2 + 0·4]] = [[3, 4], [1, 2]] (B swaps A's rows).
AB ≠ BA — matrix multiplication is associative and distributive but not commutative. Useful identities: (AB)^T = B^T·A^T; A·I = I·A = A.
15.3 Determinants
The determinant maps a square matrix to a scalar measuring invertibility (and volume scaling).
- 2 × 2: det [[a, b], [c, d]] = ad − bc.
- 3 × 3 (cofactor expansion along row 1): det A = a11·M11 − a12·M12 + a13·M13, where Mij is the determinant of the 2 × 2 minor left after deleting row i and column j; signs follow the checkerboard (−1)^(i+j).
Worked Example 1. A = [[2, 1, 3], [0, 4, 1], [5, 2, 0]].
- det A = 2·det[[4, 1], [2, 0]] − 1·det[[0, 1], [5, 0]] + 3·det[[0, 4], [5, 2]]
- = 2(4·0 − 1·2) − 1(0·0 − 1·5) + 3(0·2 − 4·5)
- = 2(−2) − (−5) + 3(−20) = −4 + 5 − 60 = −59.
Properties: det(AB) = det A · det B; det A^T = det A; swapping two rows flips the sign; a row of zeros or two identical rows forces det = 0; det(kA) = k^n det A for an n × n matrix.
15.4 Inverse of a Matrix
A square matrix A is invertible (non-singular) iff det A ≠ 0, and then A⁻¹ = adj(A)/det(A), where adj(A) is the transpose of the cofactor matrix. For 2 × 2 this collapses to a formula worth memorizing:
[[a, b], [c, d]]⁻¹ = (1/(ad − bc)) · [[d, −b], [−c, a]] — swap the diagonal, negate the off-diagonal.
Worked Example 2. A = [[4, 7], [2, 6]]: det A = 24 − 14 = 10 ≠ 0, so A⁻¹ = (1/10)·[[6, −7], [−2, 4]] = [[0.6, −0.7], [−0.2, 0.4]]. Verify: A·A⁻¹ = (1/10)·[[4·6 + 7·(−2), 4·(−7) + 7·4], [2·6 + 6·(−2), 2·(−7) + 6·4]] = (1/10)·[[10, 0], [0, 10]] = I ✔.
15.5 Solving Linear Systems — Cramer's Rule
The system a1x + b1y = c1, a2x + b2y = c2 is A·X = B with coefficient matrix A. If D = det A ≠ 0, Cramer's rule gives x = Dx/D and y = Dy/D, where Dx (resp. Dy) replaces the x-column (resp. y-column) of A with B.
Worked: x + 2y = 5 and 3x − y = 1.
- D = det[[1, 2], [3, −1]] = −1 − 6 = −7.
- Dx = det[[5, 2], [1, −1]] = −5 − 2 = −7 → x = (−7)/(−7) = 1.
- Dy = det[[1, 5], [3, 1]] = 1 − 15 = −14 → y = (−14)/(−7) = 2.
- Check: 1 + 2·2 = 5 ✔ and 3·1 − 2 = 1 ✔. Equivalently X = A⁻¹B. If D = 0 the system has no solution or infinitely many — never a unique one; and for large n, Gaussian elimination (O(n^3)) replaces Cramer's rule (O(n·n!)) in practice.
15.6 Adjacency Matrices — Matrices Meet Graphs
A graph with vertices v1..vn is stored as the adjacency matrix A with A[i][j] = 1 iff vi–vj is an edge; undirected graphs give symmetric A. Theorem: (A^k)[i][j] counts walks of length k from vi to vj.
Check on the triangle C3: A = [[0,1,1],[1,0,1],[1,1,0]]; then A^2 = [[2,1,1],[1,2,1],[1,1,2]]. Diagonal entries 2 = deg(vi) (each length-2 walk out-and-back uses one incident edge); off-diagonal 1 = the single length-2 walk between distinct vertices via the third one. Powers of A drive reachability, transitive closure (Warshall), Markov chains, and PageRank; graphics pipelines multiply 4 × 4 transformation matrices per vertex.
15.7 Common Mistakes
- Assuming AB = BA, or cancelling matrices (AB = AC does not imply B = C unless A is invertible).
- Forgetting cofactor signs (−1)^(i+j) in 3 × 3 expansions.
- Dividing by det A = 0 — always test the determinant before inverting.
🎯 Exam Focus
- If A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]], compute AB, BA, and (AB)^T, and verify (AB)^T = B^T·A^T.
- Evaluate det[[1, 2, 3], [4, 5, 6], [7, 8, 9]] and explain the significance of the result.
- Find the inverse of [[2, 1, 1], [1, 2, 1], [1, 1, 2]] by the adjoint method.
- Solve by Cramer's rule: 2x + y − z = 3, x − y + 2z = 1, 3x + 2y + z = 4.
- For the path graph P3 (v1–v2–v3), write A and compute A^2 and A^3. How many length-3 walks go from v1 to v2?
- Prove that a square matrix with two identical rows has determinant zero.