Module 1
Matrices
From notation to eigenvalues. Every named type of matrix, every standard operation, and every method the exam asks for — worked in full.
9 topics · 88 solved problems
Module contents
- 013 worked · 5 practice
Matrices, order and notation
What a matrix is, how its order is written, how to name an element, and when two matrices count as equal.
- 025 worked · 8 practice
Types of matrices
Every named matrix in the syllabus — row, column, null, diagonal, scalar, identity, triangular, symmetric, skew-symmetric, orthogonal, idempotent, involutory, nilpotent, periodic, Hermitian, unitary and more — each with its defining test and a solved check.
- 034 worked · 6 practice
Matrix operations
Addition, subtraction, scalar multiplication and the row-by-column product — with the conformability rules, the algebraic laws that survive and the ones that fail.
- 044 worked · 7 practice
Determinants
Evaluating 2×2 and 3×3 determinants, cofactor expansion along any row or column, Sarrus' rule, and the seven properties that turn a hard determinant into an easy one.
- 053 worked · 7 practice
Adjoint and inverse
Building the cofactor matrix, transposing it to get the adjoint, and dividing by the determinant to get the inverse — plus the Gauss–Jordan route and the properties of A⁻¹.
- 063 worked · 6 practice
Elementary operations, echelon form and rank
Row-reducing a matrix to echelon form, reading the rank off the staircase, and the minor method as a cross-check.
- 075 worked · 7 practice
Systems of linear equations
Matrix form AX = B, the consistency test by rank, and four solution methods — Cramer's rule, the inverse method, Gauss elimination and Gauss–Jordan — including homogeneous systems.
- 083 worked · 6 practice
Eigenvalues and eigenvectors
The characteristic equation, finding eigenvalues for 2×2 and 3×3 matrices, extracting eigenvectors, and the properties that let you check an answer in seconds.
- 092 worked · 4 practice
Cayley–Hamilton theorem
Every matrix satisfies its own characteristic equation — and how to use that to find inverses and reduce high powers without heavy computation.
Syllabus coverage
- Definition, order, types of matrices
- Addition, scalar multiplication and multiplication of matrices
- Transpose, symmetric and skew-symmetric matrices
- Determinants and their properties
- Minors, cofactors, adjoint and inverse
- Elementary transformations, echelon form and rank
- Solution of linear equations — Cramer's rule, matrix method, Gauss elimination
- Eigenvalues, eigenvectors and the Cayley–Hamilton theorem