Vectors and Matrices
Core ideas
In vectors, candidates must add, subtract and scale column vectors, find magnitudes using Pythagoras in two and three dimensions, and use position vectors to find lengths and midpoints. The scalar (dot) product helps test perpendicularity: a·b = 0 means perpendicular. Matrices require addition, subtraction, multiplication (non-commutative), finding the determinant and inverse of a 2×2 matrix, and using matrices to solve simultaneous equations. The inverse of a 2×2 matrix with entries a, b, c, d is (1/(ad − bc)) times the matrix with a and d swapped and b, c negated.
Traps
- Multiplying matrices in the wrong order.
- Forgetting that a singular matrix (determinant zero) has no inverse.
- Sign errors in the inverse formula.
- Adding vectors head-to-head rather than head-to-tail.
- Forgetting to check that the matrix product is defined (inner dimensions match).
Worked example
For A = [[3, 1], [2, 4]], determinant = 12 − 2 = 10. Then A⁻¹ = (1/10)[[4, −1], [−2, 3]].
Rule of thumb
Before inverting a 2×2 matrix, compute the determinant first; if it is zero, stop - the matrix is singular and the system may have no unique solution.