1. Computing Determinants
2x2: det([a b; c d]) = ad - bc
Cofactor expansion: det(A) = sum of a_{ij} * C_{ij} along any row or column. C_{ij} = (-1)^{i+j} M_{ij}.
Triangular matrix: det = product of diagonal entries. Choose the row/column with the most zeros for efficiency.
2. Row Operations and Determinants
| Operation | Effect on det |
|---|
| Row swap | Negates det |
| Scale row by c | Multiplies det by c |
| Row replacement | No change |
3. Key Properties
det(AB) = det(A)det(B). det(A^T) = det(A). det(A^{-1}) = 1/det(A). det(cA) = c^n det(A).
det = 0 if and only if A is singular (not invertible). Also det = 0 if any row/column is all zeros, or if two rows/columns are identical or proportional.
4. Cramer's Rule
x_i = det(A_i) / det(A), where A_i = A with column i replaced by b.
Requires det(A) nonzero. Best for small systems or single-variable solutions.
5. Geometric Interpretation
|det(A)| = area (2D) or volume (3D) of parallelogram/parallelepiped spanned by columns. Sign = orientation. det = 0 = degenerate (dependent columns).