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[Merged by Bors] - feat(LinearAlgebra/Matrix/Adjugate): M.det = 0 if M *ᵥ v = 0 where v contains a non-zero-divisor #39642

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[Merged by Bors] - feat(LinearAlgebra/Matrix/Adjugate): M.det = 0 if M *ᵥ v = 0 where v contains a non-zero-divisor #39642

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5 changes: 5 additions & 0 deletions Mathlib/LinearAlgebra/Matrix/Adjugate.lean
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Expand Up @@ -284,6 +284,11 @@ divides `b`. -/
theorem mulVec_cramer (A : Matrix n n α) (b : n → α) : A *ᵥ cramer A b = A.det • b := by
rw [cramer_eq_adjugate_mulVec, mulVec_mulVec, mul_adjugate, smul_mulVec, one_mulVec]

theorem det_eq_zero_of_mulVec_eq_zero_of_mem_nonZeroDivisors {M : Matrix n n α} {v : n → α}
(h : M *ᵥ v = 0) {i : n} (hi : v i ∈ nonZeroDivisors α) : M.det = 0 := by
apply mul_right_mem_nonZeroDivisors_eq_zero_iff hi |>.mp
simpa [adjugate_mul, smul_mulVec] using congr((M.adjugate *ᵥ $h) i)

theorem adjugate_subsingleton [Subsingleton n] (A : Matrix n n α) : adjugate A = 1 := by
ext i j
simp [Subsingleton.elim i j, adjugate_apply, det_eq_elem_of_subsingleton _ i, one_apply]
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