let n be Nat; :: thesis: for K being Field
for M being Matrix of n,K st M is invertible holds
for i, j being Nat st [i,j] in Indices (M ~) holds
(M ~) * (i,j) = (((Det M) ") * ((power K) . ((- (1_ K)),(i + j)))) * (Minor (M,j,i))
let K be Field; :: thesis: for M being Matrix of n,K st M is invertible holds
for i, j being Nat st [i,j] in Indices (M ~) holds
(M ~) * (i,j) = (((Det M) ") * ((power K) . ((- (1_ K)),(i + j)))) * (Minor (M,j,i))
let M be Matrix of n,K; :: thesis: ( M is invertible implies for i, j being Nat st [i,j] in Indices (M ~) holds
(M ~) * (i,j) = (((Det M) ") * ((power K) . ((- (1_ K)),(i + j)))) * (Minor (M,j,i)) )
assume M is invertible ; :: thesis: for i, j being Nat st [i,j] in Indices (M ~) holds
(M ~) * (i,j) = (((Det M) ") * ((power K) . ((- (1_ K)),(i + j)))) * (Minor (M,j,i))
then A1: Det M <> 0. K by Th34;
set D = Det M;
set COF = Matrix_of_Cofactor M;
let i, j be Nat; :: thesis: ( [i,j] in Indices (M ~) implies (M ~) * (i,j) = (((Det M) ") * ((power K) . ((- (1_ K)),(i + j)))) * (Minor (M,j,i)) )
assume [i,j] in Indices (M ~) ; :: thesis: (M ~) * (i,j) = (((Det M) ") * ((power K) . ((- (1_ K)),(i + j)))) * (Minor (M,j,i))
then A2: [i,j] in Indices ((Matrix_of_Cofactor M) @) by MATRIX_0:26;
then A3: [j,i] in Indices (Matrix_of_Cofactor M) by MATRIX_0:def 6;
thus (M ~) * (i,j) = (((Det M) ") * ((Matrix_of_Cofactor M) @)) * (i,j) by A1, Th35
.= ((Det M) ") * (((Matrix_of_Cofactor M) @) * (i,j)) by A2, MATRIX_3:def 5
.= ((Det M) ") * ((Matrix_of_Cofactor M) * (j,i)) by A3, MATRIX_0:def 6
.= ((Det M) ") * (Cofactor (M,j,i)) by A3, Def6
.= (((Det M) ") * ((power K) . ((- (1_ K)),(i + j)))) * (Minor (M,j,i)) by GROUP_1:def 3 ; :: thesis: verum