Matrices and Determinants
Matrices and Determinants
The exam checks matrix algebra, symmetry, invertibility through the adjoint, determinant properties and solving linear systems with inverse matrices or Cramer's rule.
Core rules
- Matrices add only with the same order; AB requires columns of A = rows of B, and generally AB ≠ BA.
- Transpose rules: (AB)ᵀ = BᵀAᵀ. A is symmetric if Aᵀ = A and skew-symmetric if Aᵀ = −A, whose diagonal entries must be zero.
- For square A of order n: |kA| = kⁿ|A|, |AB| = |A||B|, |Aᵀ| = |A|. If |A| = 0, A is singular and A⁻¹ does not exist.
- A⁻¹ = adj(A)/|A|, with A·adj A = |A|I and |adj A| = |A|ⁿ⁻¹.
- AX = B has a unique solution when |A| ≠ 0, namely X = A⁻¹B.
Worked example
For A = [[2, 3], [1, 2]], find A⁻¹.
|A| = (2)(2) − (3)(1) = 1, so A is invertible. The adjoint is [[2, −3], [−1, 2]]: swap the diagonal entries and negate the off-diagonal entries. Therefore A⁻¹ = [[2, −3], [−1, 2]]/1. Check: [[2, 3], [1, 2]]·[[2, −3], [−1, 2]] = [[1, 0], [0, 1]].
Traps
- Writing (AB)⁻¹ = A⁻¹B⁻¹; the correct order is B⁻¹A⁻¹.
- For 3×3 matrices, |2A| = 2³|A| = 8|A|, not 2|A|.
- Sign errors in cofactors when building the adjoint; the adjoint is the transpose of the cofactor matrix.
- Applying Cramer's rule when |A| = 0; the system may be inconsistent or have infinitely many solutions.
- Treating AX = B and XA = B as the same equation.