Matrices and linear transformations
Matrices and linear transformations tests matrix arithmetic, the determinant and inverse of 2 × 2 matrices, the solution of linear systems, and the geometric interpretation of matrices as transformations of the plane, including invariant points and lines.
Arithmetic, determinant and inverse
Matrix addition is elementwise, but multiplication requires matching dimensions and is not commutative, so AB is generally not BA. For a 2 × 2 matrix with entries a, b in the first row and c, d in the second, the determinant is ad − bc. If the determinant is nonzero, the inverse is (1/(ad − bc)) times the matrix with a and d swapped and b and c negated. A zero determinant means the matrix is singular and has no inverse.
Transformations and invariants
A 2 × 2 matrix acts on position vectors, producing rotations, reflections, enlargements and shears. The columns of the matrix give the images of the unit vectors (1, 0) and (0, 1). The determinant is the area scale factor; a negative determinant indicates a change of orientation. Invariant points satisfy Mv = v, found by solving (M − I)v = 0. Invariant lines through the origin satisfy Mv = λv, so the line is mapped to itself, not necessarily pointwise fixed.
Solving systems
Simultaneous equations can be written as a matrix equation and solved by multiplying by the inverse, provided the determinant is nonzero. If the determinant is zero the system has either no solution or infinitely many.
Worked example
Find the inverse of M = [[3, 1], [2, 4]].
The determinant is 3 × 4 − 1 × 2 = 10. Swapping the leading diagonal and negating the other entries gives [[4, −1], [−2, 3]], so M⁻¹ = (1/10)[[4, −1], [−2, 3]].
Traps that cost marks
- Assuming AB = BA when multiplying matrices.
- Forgetting to swap the leading diagonal entries when forming the inverse.
- Losing the factor 1/determinant, or dividing by zero for a singular matrix.
- Using Mv = λv for invariant lines and Mv = v for invariant points interchangeably.