invertibleOfDetInvertible — Mathlib

English

The inverse of a matrix A equals det(A)^{-1} times the adjugate of A when A is invertible.

Русский

Обратная матрицы A равна det(A)^{-1} умноженной на приспособление adj(A), когда A обратима.

LaTeX

$$$A^{-1} = \det(A)^{-1} \cdot \operatorname{adj}(A)$$$

Lean4

/-- If `A.det` has a constructive inverse, produce one for `A`. -/
def invertibleOfDetInvertible [Invertible A.det] : Invertible A
    where
  invOf := ⅟A.det • A.adjugate
  mul_invOf_self := by rw [mul_smul_comm, mul_adjugate, smul_smul, invOf_mul_self, one_smul]
  invOf_mul_self := by rw [smul_mul_assoc, adjugate_mul, smul_smul, invOf_mul_self, one_smul]

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