// Article · Linear algebra

Learning Matrices and Determinants: A 7-Step Roadmap from High School and Entrance Exams to Engineering Mathematics

Many students feel that matrices are something they can compute but cannot see, because the topics are taught as separate formulas. This article puts everything on one path, from multiplying matrices at school to the methods circuit simulators use to solve millions of equations, and shows how far each goal needs to go.

By Ajarn ModUpdated 25 Oct 2026

The 30-second summary

Every topic answers one question: how to solve $A\mathbf x = \mathbf b$ , whether a solution exists, and how far it can be trusted
Entrance exams: steps 1–3 (matrices, determinants, inverse, Cramer) · university: add steps 4–5 (row reduction, rank) · engineering: steps 6–7
Practise with the 45 exam problems and random drill and check homework with the matrix lab (in Thai)

Why matrices: systems of equations are everywhere

A three-mesh circuit gives three current equations, a steel roof truss gives force-balance equations at every joint, and a heat model gives one equation per point. All of them fit in one short line:

A\mathbf x = \mathbf b

Here $A$ holds the coefficients, $\mathbf x$ the unknowns and $\mathbf b$ the given values. A matrix is therefore not just a table of numbers but a language that lets you handle hundreds of equations as if they were one.

The 7-step map

The path from school to engineering; colour shows the level: orange = entrance exam · blue = undergraduate · green = engineering practice.

Step	Answers	Lesson (in Thai)
1 Matrices	how to write a system as $A\mathbf x = \mathbf b$ and multiply	7.1
2 Determinants	is the solution unique ( $\det A \ne 0$ )?	7.2
3 Inverse · Cramer	what is the solution (small systems)?	7.4 · 7.5
4 Row reduction	solve every case, even when $\det = 0$	7.3
5 Rank	one, infinitely many or no solutions?	7.10
6 Numerical methods	solve large systems fast and accurately	7.11
7 Eigenvalues · applications	how does the system respond (poles, modes)?	7.6 · 7.7

If your goal is a university entrance exam

Multiply matrices fluently, and remember $AB \ne BA$ but $(AB)^T = B^TA^T$
2×2 and 3×3 determinants (Sarrus or cofactors)
All determinant properties, especially $\det(kA) = k^n\det A$ and $\det(\operatorname{adj}A) = (\det A)^{n-1}$ — summarised in the 12 determinant properties article
The 2×2 inverse and matrix equations $AX = B$ (multiply by $A^{-1}$ on the left)
Cramer's rule for one unknown, and systems with a parameter ( $\Delta = 0$ : no solution versus infinitely many) — see the Cramer's rule article

If you study engineering mathematics at university

Gaussian and Gauss–Jordan elimination, used for solutions, inverses and rank
Rank and the Rouché–Capelli theorem, “rank for every $k$ ” problems, the null space — see the rank of a matrix article
Eigenvalues, eigenvectors and diagonalisation, which lead into control systems and differential equations
Numerical methods: LU, Jacobi, Gauss–Seidel and the condition number — see the numerical methods comparison

A 7-day practice plan (45–60 minutes a day)

Day 1: read matrices and use the cell-by-cell product viewer
Day 2: determinants up to the exam shortcuts; do set A, problems 1–6
Day 3: inverse and Cramer's rule; set A, problems 7–12
Day 4: row reduction; set A, problems 13–18
Day 5: rank and the three-planes lab; set B, problems 1–8
Day 6: numerical methods; set C
Day 7: the random drill at every level until you get 10 in a row; revisit the formula codes in the solutions you missed

Continue in the library (in Thai)

Frequently asked questions

Where should I start learning matrices?

Start with size, addition, multiplication and the transpose, then 2×2 and 3×3 determinants and their properties, and only then the inverse, Cramer's rule and row reduction for solving systems. Every later topic relies on matrix multiplication and determinants.

How much matrix work do entrance exams need?

Fluent multiplication and transposes, 2×2 and 3×3 determinants, determinant properties (det(kA), det(AB), det(A⁻¹), det(adj A)), the 2×2 inverse, matrix equations, Cramer's rule and systems with a parameter. Rank and numerical methods belong to university courses.

What do engineers use matrices for?

To solve the equations of electric circuits (nodal and mesh analysis), power systems, structures and vibrations, to find the poles of control systems from eigenvalues, and to run numerical methods such as LU or Gauss–Seidel inside simulators with thousands to millions of unknowns.

Where can I practise matrix problems?

The exam-practice page in the library has 45 problems with step-by-step solutions at three levels (entrance exam, undergraduate and numerical methods) and a random problem generator that checks your answers instantly.