-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
lemma reverse_charpoly (M : Matrix n n R) :
    M.charpoly.reverse = M.charpolyRev

Proof state:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
a✝ : Nontrivial R
t : R[T;T⁻¹] := T 1
t_inv : R[T;T⁻¹] := T (-1)
p : R[T;T⁻¹] := ((scalar n) t - M.map ⇑LaurentPolynomial.C).det
q : R[T;T⁻¹] := (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C).det
⊢ M.charpoly.reverse = M.charpolyRev

Tactic:
have ht : t_inv * t = 1 := by rw [← T_add, neg_add_cancel, T_zero]

Result:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
a✝ : Nontrivial R
t : R[T;T⁻¹] := T 1
t_inv : R[T;T⁻¹] := T (-1)
p : R[T;T⁻¹] := ((scalar n) t - M.map ⇑LaurentPolynomial.C).det
q : R[T;T⁻¹] := (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C).det
ht : t_inv * t = 1
⊢ M.charpoly.reverse = M.charpolyRev

Proof state:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
a✝ : Nontrivial R
t : R[T;T⁻¹] := T 1
⊢ M.charpoly.reverse = M.charpolyRev

Tactic:
let t_inv : R[T;T⁻¹] := T (-1)

Result:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
a✝ : Nontrivial R
t : R[T;T⁻¹] := T 1
t_inv : R[T;T⁻¹] := T (-1)
⊢ M.charpoly.reverse = M.charpolyRev

Proof state:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
a✝ : Nontrivial R
t : R[T;T⁻¹] := T 1
t_inv : R[T;T⁻¹] := T (-1)
p : R[T;T⁻¹] := ((scalar n) t - M.map ⇑LaurentPolynomial.C).det
q : R[T;T⁻¹] := (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C).det
ht : t_inv * t = 1
hp : toLaurentAlg M.charpoly = p
hq : toLaurentAlg M.charpolyRev = q
⊢ t_inv ^ Fintype.card n * p = invert q

Tactic:
rw [← det_smul, smul_sub, scalar_apply, ← diagonal_smul, Pi.smul_def, smul_eq_mul, ht,
  diagonal_one, invert.map_det]

Result:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
a✝ : Nontrivial R
t : R[T;T⁻¹] := T 1
t_inv : R[T;T⁻¹] := T (-1)
p : R[T;T⁻¹] := ((scalar n) t - M.map ⇑LaurentPolynomial.C).det
q : R[T;T⁻¹] := (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C).det
ht : t_inv * t = 1
hp : toLaurentAlg M.charpoly = p
hq : toLaurentAlg M.charpolyRev = q
⊢ (1 - t_inv • M.map ⇑LaurentPolynomial.C).det = (invert.mapMatrix (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C)).det

Proof state:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
a✝ : Nontrivial R
t : R[T;T⁻¹] := T 1
t_inv : R[T;T⁻¹] := T (-1)
p : R[T;T⁻¹] := ((scalar n) t - M.map ⇑LaurentPolynomial.C).det
q : R[T;T⁻¹] := (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C).det
ht : t_inv * t = 1
hp : toLaurentAlg M.charpoly = p
hq : toLaurentAlg M.charpolyRev = q
⊢ M.charpoly.reverse = M.charpolyRev

Tactic:
suffices t_inv ^ Fintype.card n * p = invert q by
  apply toLaurent_injective
  rwa [toLaurent_reverse, ← coe_toLaurentAlg, hp, hq, ← involutive_invert.injective.eq_iff,
    _root_.map_mul, involutive_invert p, charpoly_natDegree_eq_dim,
    ← mul_one (Fintype.card n : ℤ), ← T_pow, map_pow, invert_T, mul_comm]

Result:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
a✝ : Nontrivial R
t : R[T;T⁻¹] := T 1
t_inv : R[T;T⁻¹] := T (-1)
p : R[T;T⁻¹] := ((scalar n) t - M.map ⇑LaurentPolynomial.C).det
q : R[T;T⁻¹] := (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C).det
ht : t_inv * t = 1
hp : toLaurentAlg M.charpoly = p
hq : toLaurentAlg M.charpolyRev = q
⊢ t_inv ^ Fintype.card n * p = invert q