Determinant formulas
2×2: det = ad − bc
3×3: a(ei − fh) − b(di − fg) + c(dh − eg)
For 4×4, the calculator uses cofactor expansion along the first row, calling the 3×3 routine recursively. For larger matrices, LU decomposition is faster — consider a numerical-linear-algebra library.
What the determinant tells you
- det = 0 — the matrix is singular: rows or columns are linearly dependent, the system Ax = b has no unique solution, the linear transformation is non-invertible.
- det > 0 — the transformation preserves orientation; the absolute value is the volume scaling factor.
- det < 0 — the transformation reverses orientation (mirror flip); |det| is still the volume scaling factor.