Theorems · Theorem · linear algebra
Matrix.det_apply
∀ {n : Type u_2} [inst : DecidableEq n] [inst_1 : Fintype n] {R : Type v} [inst_2 : CommRing R] (M : Matrix n n R),
M.det = ∑ σ, Equiv.Perm.sign σ • ∏ i, M (σ i) i- Cited by
- 16 results in Mathlib
- Foundations
- Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- DecidableEqFintypeCommRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Fintypestatement and proof · cited by 7,736
- Finset.sumstatement · cited by 5,195
- Matrixstatement and proof · cited by 4,303
- MonoidHomstatement · cited by 3,629
- Finset.univstatement · cited by 3,473
- Unitsstatement · cited by 2,804
- Finset.prodstatement · cited by 2,356
- Equiv.Permstatement · cited by 1,375
- Matrix.detstatement · cited by 665
- Equiv.Perm.signstatement · cited by 138
Cited by16
Results whose statement or proof uses this declaration.
- Matrix.det_uniqueproof · cited by 15
- Matrix.det_apply'proof · cited by 11
- Matrix.det_isEmptyproof · cited by 9
- Continuous.matrix_detproof · cited by 7
- IsIntegral.detproof · cited by 3
- Matrix.charpoly_sub_scalarproof · cited by 1
- exteriorPower.alternatingMapToDual_apply_ιMultiproof · cited by 1
- Polynomial.natDegree_det_X_add_C_leproof · cited by 1
- Algebra.discr_mul_isIntegral_mem_adjoinproof · cited by 1
- Matrix.det_succ_column_zeroproof · cited by 1
- Matrix.coeff_charpoly_mem_ideal_powproof · cited by 1
- Polynomial.coeff_det_X_add_C_cardproof · cited by 1