Determinants In-depth
linear-algebra
Definition
The determinant is a scalar value computed from a square matrix \(A \in \mathbb{R}^{n \times n}\), denoted \(\det(A)\) or \(|A|\).
2×2 Case
\[\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc\]
3×3 Case — Cofactor Expansion
Expanding along the first row:
\[\det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}\]
where the cofactor \(C_{ij} = (-1)^{i+j} M_{ij}\) and \(M_{ij}\) is the minor (determinant of the submatrix obtained by deleting row \(i\) and column \(j\)).
Properties
| Property | Statement |
|---|---|
| Transpose | \(\det(A^T) = \det(A)\) |
| Product | \(\det(AB) = \det(A)\det(B)\) |
| Inverse | \(\det(A^{-1}) = \frac{1}{\det(A)}\) |
| Singular | \(\det(A) = 0 \Leftrightarrow A\) is not invertible |
| Row swap | Swapping two rows negates the determinant |
| Row scale | Scaling a row by \(c\) scales \(\det\) by \(c\) |
Geometric Interpretation
- In \(\mathbb{R}^2\): \(|\det(A)|\) is the area of the parallelogram formed by the column vectors
- In \(\mathbb{R}^3\): \(|\det(A)|\) is the volume of the parallelepiped
- \(\det(A) < 0\) means the transformation reverses orientation
Relation to Eigenvalues
\[\det(A) = \prod_{i=1}^{n} \lambda_i\]
If any eigenvalue is zero, the matrix is singular.