Theorems · Theorem · linear algebra
Matrix.nonsing_inv_mul
∀ {n : Type u'} {α : Type v} [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : CommRing α] (A : Matrix n n α),
IsUnit A.det → A⁻¹ * A = 1The nonsingular inverse of A is a left inverse.
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FintypeDecidableEqCommRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Fintypestatement and proof · cited by 7,736
- Matrixstatement and proof · cited by 4,303
- IsUnitstatement and proof · cited by 1,602
- Matrix.detstatement and proof · cited by 665
- Invertibleproof · cited by 549
- invOf_mul_selfproof · cited by 20
- IsUnit.nonempty_invertibleproof · cited by 16
- Matrix.isUnit_iff_isUnit_detproof · cited by 12
- Matrix.invOf_eq_nonsing_invproof · cited by 11
Cited by13
Results whose statement or proof uses this declaration.
- Matrix.inv_mul_of_invertibleproof · cited by 7
- Matrix.nonsing_inv_cancel_or_zeroproof · cited by 4
- Matrix.zpow_add_oneproof · cited by 3
- Matrix.charpoly_units_conjproof · cited by 2
- Matrix.nonsing_inv_mul_cancel_rightproof · cited by 2
- Matrix.det_nonsing_inv_mul_detproof · cited by 1
- Algebra.discr_mul_isIntegral_mem_adjoinproof · cited by 1
- Matrix.nonsing_inv_mul_cancel_leftproof · cited by 1
- Matrix.invertibleOfIsUnitDetproof · cited by 0
- Matrix.zpow_neg_mul_zpow_selfproof · cited by 0
- Matrix.GeneralLinearGroup.coe_map_inv_mul_mapproof · cited by 0
- Matrix.inv_smulproof · cited by 0