Determinant¶
Parameters¶
- \(A\): square matrix
- \(\det(A)\): signed volume scaling factor
What it means¶
The determinant measures volume scaling (and orientation sign) of the linear map defined by a matrix.
What it's used for¶
- Invertibility checks.
- Change-of-variables and geometry.
Key properties¶
- \(\det(AB)=\det(A)\det(B)\).
- Determinant is zero iff columns/rows are linearly dependent.
Common gotchas¶
- Determinants are numerically unstable for large systems; use factorizations instead.
- Large determinant magnitude does not directly imply good conditioning.
Example¶
\(\det\!\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc\).
How to Compute (Pseudocode)¶
Input: square matrix A
Output: det(A)
compute an LU decomposition with row permutations: P A = L U
sign <- sign of the permutation matrix P
detA <- sign * product of diagonal entries of U
return detA
Complexity¶
- Time: \(O(n^3)\) for dense \(n\times n\) matrices using LU-type methods
- Space: \(O(n^2)\) for dense matrix/factor storage
- Assumptions: LU-based numerical computation shown; direct cofactor expansion is exponentially expensive and used only for tiny symbolic examples