81. Determinant
A scalar attached to every square matrix that measures the signed scale factor the matrix applies to areas / volumes when used as a linear map.
Notation: , or .
81.1. From first principles: the case
Take two vectors :
They span a parallelogram. Its signed area is:
That signed area is exactly the determinant of the matrix whose columns are :
Sign:
- if is counterclockwise from (preserved orientation)
- if clockwise (orientation flipped)
- if and are collinear (parallelogram degenerates)
Example
The unit square maps to a parallelogram of area — six times bigger.
The columns are parallel (one is twice the other) — image collapses to a line, area .
81.2. : signed volume of the parallelepiped
Three column vectors span a parallelepiped. Its signed volume is:
This is the cofactor expansion along the first column — three determinants, alternating sign.
81.3. General : cofactor expansion
For an matrix , expand along any row (or any column ):
where is the minor — the determinant of the submatrix obtained by deleting row and column of (see Minor).
The factor produces the checkerboard sign pattern:
The recursion bottoms out at the case: .
81.4. Key properties
- Identity:
- Multiplicativity:
- Transpose:
- Inverse: (when defined)
- Scalar multiple: for an matrix
- Triangular matrix: product of diagonal entries
- Row swap: flips the sign
- Row scaled by : multiplies by
- Row multiple of another row: leaves unchanged (basis of Gaussian-elimination computation)
81.5. What tells you about
| Implication | |
|---|---|
| is invertible (see Matrix Inverse); columns are linearly independent (see Linear Independence); columns span ; has full rank (see Rank); has a unique solution for every . | |
| is singular — non-invertible; columns are linearly dependent; image is a strict subspace; has non-trivial solutions; the linear map collapses dimension. | |
81.6. Geometric interpretation
For a linear map given by matrix :
- = factor by which scales -dimensional volume
- = whether preserves or reverses orientation
Example
Reflection across the -axis:
Areas preserved (factor ); orientation reversed (sign).
81.7. Computing in practice
- / : direct formula above.
- Larger matrices: cofactor expansion is — impractical past .
- Standard method: row-reduce to triangular form, multiply diagonal entries, track row-operation effects on (see Gaussian Elimination). This is .
- LU decomposition (see LU Decomposition): of ‘s diagonal entries (since has unit diagonal).
81.8. Connection to other concepts
- Minor — building block for cofactor expansion
- Adjugate — matrix of cofactors, used in
- Eigenvectors & Eigenvalues — eigenvalues are roots of
- Unimodularity — matrices with