Basic Mathematical Applications
Vectors, matrices, statistics and probability
Basic Mathematical Applications tests the applied tools of vectors and matrices and the interpretation of data through statistics and probability. Questions are usually short but precise: a matrix must be multiplied in the correct order, and a probability answer must match the sample space.
Core rules
- Vectors have magnitude and direction. Add vectors component-wise; multiply by a scalar to change length. The magnitude of (x, y) is sqrt(x^2 + y^2). A position vector is measured from the origin.
- Matrices can be added only when they have the same order. For AB, the number of columns of A must equal the number of rows of B; matrix multiplication is not commutative.
- For A = [[a, b], [c, d]], det A = ad - bc. If det A is not zero, A-inverse = (1/(ad - bc))[[d, -b], [-c, a]]. Solve AX = B as X = A-inverse B.
- Mean = sum of x / n for raw data; median is the middle value after ordering; mode is the most frequent. Range = largest - smallest; interquartile range = Q3 - Q1.
- P(A or B) = P(A) + P(B) - P(A and B). For mutually exclusive events, P(A and B) = 0. For independent events, P(A and B) = P(A)P(B).
Worked example
Solve 2x + y = 5 and x + 3y = 10 using matrices.
Write A = [[2, 1], [1, 3]], X = [[x], [y]], B = [[5], [10]]. Then det A = (2)(3) - (1)(1) = 5. The inverse is (1/5)[[3, -1], [-1, 2]]. So X = A-inverse B = (1/5)[[3, -1], [-1, 2]][[5], [10]] = (1/5)[[15 - 10], [-5 + 20]] = [[1], [3]]. Hence x = 1, y = 3. Substitute into both equations to check.
Traps
Do not multiply matrices in the wrong order; AB and BA are usually different. An inverse exists only for a square matrix with nonzero determinant. In probability, do not add probabilities for events that are not mutually exclusive without subtracting the overlap. Do not treat independent events as mutually exclusive - they are different ideas. For grouped data, use class midpoints for the mean and remember that the median lies in a class interval, not necessarily at a data value.