Matrices and Determinants
Questions here test matrix algebra, the meaning of a determinant, inverse of a square matrix, and solving a small linear system by matrix or Cramer's rule.
Matrix rules
- Equality requires the same order and equal corresponding entries. AB is defined only when columns of A equal rows of B; generally AB ≠ BA.
- A is symmetric if A′ = A, skew-symmetric if A′ = −A (all diagonal entries then zero).
- For square matrices of order n: |A′| = |A|, |kA| = kⁿ|A|, |AB| = |A||B|, and A is singular when |A| = 0.
- Inverse: A⁻¹ = adj A / |A|, where adj A is the transpose of the cofactor matrix; (AB)⁻¹ = B⁻¹A⁻¹.
Worked example
Take A = [[2,3],[1,4]]. Then |A| = 2(4) − 3(1) = 5 ≠ 0, so A⁻¹ exists. For a 2×2 matrix, adj A is obtained by swapping the diagonal entries and negating the off-diagonal ones: adj A = [[4,−3],[−1,2]]. Hence A⁻¹ = (1/5)[[4,−3],[−1,2]]. Check: [[2,3],[1,4]] · [[4,−3],[−1,2]] = [[5,0],[0,5]] = 5I, so dividing by 5 gives I. For a system AX = B with |A| ≠ 0, X = A⁻¹B.
Traps that cost marks
- Writing |kA| = k|A| instead of kⁿ|A|.
- Expanding (A+B)² as A² + 2AB + B² - invalid unless AB = BA.
- Forgetting the sign pattern (−1)^(i+j) in cofactors, or forgetting to transpose when forming adj A.
- Assuming AB = 0 forces A = 0 or B = 0; it does not.
- Inverting a singular matrix, or mixing rows and columns in multiplication.