Matrices, determinants and linear transformations

Learning objectives

Carry out addition, subtraction, scalar multiplication and multiplication of matrices of conformable order, and use the facts that matrix multiplication is associative and distributive but in general not commutative

Evaluate the determinant of a 2×2 and of a 3×3 matrix, find the inverse of a non-singular matrix, and state that a matrix has an inverse exactly when its determinant is non-zero

(Paper 2) Write a system of three linear equations as Ax = b, solve it using A⁻¹ when A is non-singular, and interpret a zero determinant as meaning the system is either inconsistent or has infinitely many solutions

Represent reflections, rotations, enlargements, stretches and shears of the plane by 2×2 matrices, and obtain the matrix of a composite transformation as a product taken in the correct order

Interpret the determinant as the area scale factor of a plane transformation, with a negative value showing that orientation is reversed

Find the invariant points and the invariant lines of a linear transformation of the plane, distinguishing a line of invariant points, on which every point is fixed, from an invariant line, which maps onto itself while its points move along it