Matrices and Determinants
Matrices and Determinants is the first algebra unit: the exam tests whether you can combine matrices, compute determinants and inverses, and solve a two- or three-variable system without arithmetic slips.
Core rules
- A matrix of order m×n has m rows and n columns. Addition needs identical orders; multiplication AB needs columns of A = rows of B, and AB ≠ BA in general.
- For A = [[a,b],[c,d]], det A = ad − bc. A square matrix is singular when det A = 0 and has no inverse.
- Inverse: A⁻¹ = adj A / det A, where adj A is the transpose of the cofactor matrix. For a 2×2 matrix, swap a and d, change signs of b and c, then divide by the determinant.
- Cramer's rule for ax+by=e, cx+dy=f: D = ad−bc, Dx = ed−bf, Dy = af−ec, so x = Dx/D and y = Dy/D, provided D ≠ 0.
- A homogeneous system has a non-trivial solution only when the coefficient determinant is zero.
Worked example
Solve 2x + 3y = 8 and 5x − y = 3. D = (2)(−1) − (3)(5) = −2 − 15 = −17. Dx = (8)(−1) − (3)(3) = −8 − 9 = −17, so x = −17/−17 = 1. Dy = (2)(3) − (8)(5) = 6 − 40 = −34, so y = −34/−17 = 2. Check: 2(1)+3(2)=8 and 5(1)−2=3.
Traps
Expanding a 3×3 determinant along a row but forgetting the alternating + − + signs; writing adj A without transposing the cofactor matrix; using Cramer's rule when D = 0; and assuming AB = BA, which changes answers in product and inverse problems.