Matrices and Determinants
Section A (compulsory) and Mathematics Unit II; also Applied Mathematics Unit II.
About this unit
Matrices are rectangular arrays of numbers that store data compactly and let us handle many linear equations at once. Determinants attach a single number to every square matrix and tell us whether that matrix can be inverted. In this unit you learn the types of matrices, the rules for adding and multiplying them, how to evaluate determinants up to order 3, how to find the adjoint and inverse, and how to solve systems of linear equations with the matrix method. These tools are used in economics, computer graphics, statistics and linear programming.
Key ideas
- Order m × n means m rows and n columns.
- Matrix multiplication AB needs the columns of A to equal the rows of B, and in general AB ≠ BA.
- Transpose, symmetric (Aᵀ = A) and skew-symmetric (Aᵀ = −A) matrices.
- Determinants, minors and cofactors; area of a triangle by determinants.
- A⁻¹ = (1/|A|) adj A, which exists only when |A| ≠ 0.
- A system AX = B has a unique solution X = A⁻¹B when |A| ≠ 0.
How to prepare
Practise 2 × 2 and 3 × 3 multiplication and determinant expansion until it is quick and error-free. Memorise the properties |AB| = |A||B|, |kA| = kⁿ|A| and |adj A| = |A|ⁿ⁻¹. Expect numericals on inverses and on the consistency of equations.
Topics in this chapter
Matrices and their types
Order, equality, zero, square, diagonal, identity, symmetric and skew-symmetric matrices; transpose.
5 questionsAlgebra of matrices
Addition, scalar multiplication and multiplication, their properties and non-commutativity.
5 questionsDeterminants
Determinants up to 3 × 3, properties, minors, cofactors and area of a triangle.
6 questionsAdjoint and inverse of a matrix
Invertible matrices, adjoint and inverse; (AB)⁻¹ = B⁻¹A⁻¹.
5 questionsSolving systems of linear equations
Consistency, number of solutions and solving equations in two or three variables by the matrix method.
5 questions