Matrices & Determinants

Matrices organise systems of numbers and linear maps. Learn matrix types (square, diagonal, identity, symmetric, triangular, orthogonal), addition and multiplication, transpose and conjugate transpose, determinants by cofactor expansion and row reduction, minors and cofactors, the inverse and adjugate, and properties like det(AB) = det A det B.

Circuit equations, truss forces, robot transformations, image processing and finite element models are all matrices. The determinant tells whether a system has a unique solution and how a transformation scales area or volume.

An analogy

A matrix is a machine that transforms vectors: it can stretch, rotate, shear or flatten space. The determinant is the machine's area-scaling factor — zero means it flattens space and information is lost.

Matrix operations

Add entry-by-entry (same size). Multiply row-by-column; order matters. The identity I leaves vectors unchanged. Transpose swaps rows and columns.

Special matrices

Symmetric (A = Aᵀ: stiffness, covariance), diagonal, triangular (easy determinants: product of diagonal), orthogonal (QᵀQ = I: rotations), sparse (FEM), Hermitian (complex, A = Aᴴ).

Determinants

Cofactor expansion is fine for small matrices; row reduction is efficient for large ones (row swaps flip sign, scaling a row scales det). det = 0 ⇔ singular ⇔ columns linearly dependent.

Geometric meaning

|det A| is the volume scaling of the transformation; its sign indicates orientation reversal. Computationally, never invert to solve Ax = b — factorise instead.

Formulas

• Product: (AB)ᵢⱼ = Σₖ aᵢₖ bₖⱼ (columns of A = rows of B); AB ≠ BA in general
• 2 × 2 determinant & inverse: det [[a, b], [c, d]] = ad − bc; A⁻¹ = (1/det)[[d, −b], [−c, a]]
• Cofactor expansion: det A = Σⱼ aᵢⱼ Cᵢⱼ (any row i)
• Inverse: A⁻¹ = adj(A)/det A (det A ≠ 0)
• Properties: det(AB) = det A det B; det Aᵀ = det A; (AB)⁻¹ = B⁻¹A⁻¹; (AB)ᵀ = BᵀAᵀ

Worked example: 2 × 2 inverse

Given: A = [[4, 7], [2, 6]]

Find: det and A⁻¹.

• det = 24 − 14 = 10
• A⁻¹ = (1/10)[[6, −7], [−2, 4]]

Answer: [[0.6, −0.7], [−0.2, 0.4]]

Worked example: 3 × 3 determinant

Given: [[1, 2, 3], [0, 4, 5], [1, 0, 6]]

Find: det.

• Expand row 1: 1(24 − 0) − 2(0 − 5) + 3(0 − 4)
• = 24 + 10 − 12

Answer: 22

Common mistakes

• Wrong: Assuming AB = BA. — Right: Matrix multiplication is not commutative.
• Wrong: (AB)⁻¹ = A⁻¹B⁻¹. — Right: (AB)⁻¹ = B⁻¹A⁻¹.

Applications

• Structures: Stiffness matrices K u = F.
• Robotics & graphics: Rotation and homogeneous transforms.
• Circuits: Nodal and mesh matrices.

Key takeaways

• Row × column; AB ≠ BA.
• det via cofactors or row reduction.
• det = 0 ⇔ singular.
• Orthogonal matrices preserve lengths.

Going further

• Block matrices: Partitioned operations for large systems.
• Kronecker products: Structured matrices in signal processing.
• Sparse storage: CSR/CSC formats in FEM.