Matrices and Determinants
Matrices and Determinants tests matrix arithmetic, properties of determinants, the adjoint and inverse, and the solution of linear systems. The chapter rewards precision: most errors in the exam are sign or order errors rather than conceptual ones.
Matrix and determinant rules
Addition requires identical orders. A product AB needs the number of columns of A to equal the number of rows of B, and generally AB ≠ BA. Transposes reverse products, (AB)ᵀ = BᵀAᵀ, and so do inverses, (AB)⁻¹ = B⁻¹A⁻¹. For determinants, |AB| = |A||B|, |Aᵀ| = |A|, and |kA| = kⁿ|A| for an n × n matrix. Swapping two rows changes the sign of the determinant, and two identical rows give zero. When |A| = 0 the matrix is singular; otherwise A⁻¹ = adj A/|A|, where adj A is the transpose of the cofactor matrix. Cramer's rule gives x = D₁/D, y = D₂/D, z = D₃/D provided D ≠ 0.
Worked example
Solve 2x + 3y = 7 and x − y = 1. Here D = (2)(−1) − (3)(1) = −5. Then D_x = (7)(−1) − (3)(1) = −10, so x = (−10)/(−5) = 2. Also D_y = (2)(1) − (7)(1) = −5, so y = 1. Checking: 2(2) + 3(1) = 7, correct.
Traps
- Writing A⁻¹ = |A|/adj A instead of adj A/|A|.
- Reversing the order in the product rule for inverses.
- Forgetting to transpose the cofactor matrix when forming the adjoint.
- Applying |kA| = k|A| instead of kⁿ|A|.