Theorems · Theorem · linear algebra

Matrix.mulVec_cramer

• #97 of the 100 theorems: Cramer’s Rule

∀ {n : Type v} {α : Type w} [inst : DecidableEq n] [inst_1 : Fintype n] [inst_2 : CommRing α] (A : Matrix n n α)
  (b : n → α), A.mulVec (A.cramer b) = A.det • b

A stronger form of Cramer's rule that allows us to solve some instances of A * x = b even if the determinant is not a unit. A sufficient (but still not necessary) condition is that A.det divides b.

Defined in

Mathlib.LinearAlgebra.Matrix.Adjugate

Cited by

2 results in Mathlib

Foundations

Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DecidableEqFintypeCommRing