Matrices for Students: Adding, Multiplying, Determinants
May 9, 2026 · 5 min · matrices · matrix math · linear algebra basics
A matrix is just a rectangular grid of numbers. They're useful because lots of real problems happen in grids — solving systems of equations, computer graphics, statistics.
Addition and subtraction
Matrices must be the same size. Add or subtract corresponding entries.
[1 2] [5 6] [6 8]
[3 4] + [7 8] = [10 12]
Multiplication
Different from element-wise multiplication. To multiply matrix A by matrix B, the number of columns in A must equal the number of rows in B.
The (i, j) entry of the product is the dot product of row i of A and column j of B.
This sounds complicated. After three worked examples it becomes mechanical.
Determinant of a 2×2
For [[a, b], [c, d]] the determinant is ad - bc.
The determinant tells you whether the matrix is invertible (non-zero determinant means it is) and whether a system of equations has a unique solution.
Inverse of a 2×2
If det(A) ≠ 0:
A⁻¹ = (1 / det(A)) × [[d, -b], [-c, a]]
Where you'll meet matrices
- A Level Further Maths
- IB Math AA HL
- AP Calculus BC (rare)
- First-year university CS, engineering, economics
Common pitfalls
- Trying to multiply incompatible matrices
- Confusing element-wise with matrix multiplication
- Wrong sign in the inverse formula
- Computing the determinant of a non-square matrix (only square matrices have one)