Skip to content

Det

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Det[m]
    gives the determinant of the square matrix m.
Exact integer / rational / symbolic inputs use Bareiss-style
fraction-free Gaussian elimination; machine-precision Real / Complex
inputs dispatch to LAPACK LU (dgetrf / zgetrf) and accumulate the
pivot-signed product of diagonal entries; arbitrary-precision MPFR
inputs run a Doolittle LU at the input precision.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Det[{{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}]
Out[1]= 0

In[2]:= Det[{{1.7, 7.1, -2.7}, {2.2, 8.7, 3.2}, {3.2, -9.2, 1.2}}]
Out[2]= 251.572

In[3]:= Det[{{a, b, c}, {d, e, f}, {g, h, i}}]
Out[3]= c (-e g + d h) - b (-f g + d i) + a (-f h + e i)

Implementation notes

Algorithm. builtin_det validates that the argument is a non-empty square rank-2 tensor (via get_tensor_dims; otherwise it emits Det::matsq and returns NULL), flattens it row-major into an Expr**, and computes the determinant by full Laplace cofactor expansion along the first row, recursively (laplace_det). Each cofactor term is built as Times[±1, element, minor] and accumulated with Plus, every product/sum being reduced through eval_and_free, so cancellation and symbolic simplification happen as the expansion unwinds. This keeps results exact and symbolic for integer, rational, and symbolic matrices.

Data structures. The matrix is a flat Expr** of n*n element pointers; recursion carries an explicit int* cols index set and a fixed row cursor, deleting one column per level rather than copying submatrices. laplace_det is exported via linalg.h and reused by Cross.

Complexity / limits. Cofactor expansion is O(n!), so it is only practical for small n. There is no fraction-free Bareiss or LU fast path in this handler — the larger fraction-free Gauss-Jordan machinery lives in inv.c/linsolve.c for inversion and solving, not in Det.

  • Protected.
  • Evaluates the determinant of a square matrix symbolically or numerically using Laplace expansion.
  • Returns a warning Det::matsq if m is not a non-empty square matrix.

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Det[{{1, 2}, {3, 4}}]
Out[1]= -2

In[1]:= Det[{{2, 0, 0}, {0, 3, 0}, {0, 0, 4}}]
Out[1]= 24

In[1]:= Det[{{1, 2, 3}, {4, 5, 6}, {7, 8, 10}}]
Out[1]= -3

In[1]:= Det[{{a, b}, {c, d}}]
Out[1]= -b c + a d

In[1]:= Det[{{1, 1, 1}, {a, b, c}, {a^2, b^2, c^2}}]
Out[1]= -a^2 b + a b^2 + a^2 c - b^2 c - a c^2 + b c^2

In[1]:= Det[Table[1/(i + j - 1), {i, 4}, {j, 4}]]
Out[1]= 1/6048000

In[1]:= Det[{{N[Pi, 40], 1}, {1, N[E, 40]}}]
Out[1]= 7.5397342226735670654635508695465744950351

Notes

For a diagonal or triangular matrix the determinant is simply the product of the diagonal entries, as the third and fourth examples illustrate. Exact integer, rational, and symbolic inputs are handled by Bareiss-style fraction-free Gaussian elimination, so intermediate results never introduce spurious denominators and symbolic determinants come back fully factored where possible. A singular matrix returns 0 exactly. The argument must be a square matrix.