Matrices, determinants and solving linear systems
A matrix is a rectangular arrangement of numbers in rows and columns; an m × n matrix has m rows and n columns. Matrices of the same order are added element by element. A product AB exists only when the number of columns of A equals the number of rows of B, and each entry is found by multiplying a row of A with a column of B and adding the results. Matrix multiplication is generally not commutative, so AB and BA may differ. The transpose Aᵀ swaps rows and columns, and a matrix is symmetric when Aᵀ = A.
Every square matrix has a determinant, a single number calculated from its entries. For a 2 × 2 matrix with rows (a, b) and (c, d), the determinant is ad − bc, and a 3 × 3 determinant is expanded along a row using minors and cofactors. A matrix has an inverse only if its determinant is non-zero, and then A⁻¹ = adj A ÷ |A|, where adj A is the transpose of the matrix of cofactors. A system of linear equations written as AX = B then has the unique solution X = A⁻¹B.
Why it matters
Matrices store and transform data efficiently, which is why they are used in computer graphics to rotate and scale images, in cryptography to encode messages, in economics for input–output models and in machine learning. Determinants show quickly whether a system of equations has a unique solution and also give the area of a triangle from its vertices. This unit carries steady marks in the Class 12 board paper.
Key terms
- Order of a matrix
- The number of rows and columns of a matrix, written as m × n.
- Determinant
- A single number calculated from a square matrix that shows whether it can be inverted.
- Cofactor
- A signed minor, used to expand determinants and to build the adjoint matrix.
- Inverse matrix
- The matrix A⁻¹ that multiplies A to give the identity matrix.