let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 = M1 * M2 & M1 is invertible holds
M1 commutes_with M2
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 = M1 * M2 & M1 is invertible holds
M1 commutes_with M2
let M1, M2 be Matrix of n,K; :: thesis: ( M1 = M1 * M2 & M1 is invertible implies M1 commutes_with M2 )
assume that
A1: M1 = M1 * M2 and
A2: M1 is invertible ; :: thesis: M1 commutes_with M2
A3: M1 ~ is_reverse_of M1 by A2, Def4;
A4: ( len M2 = n & width (M1 ~) = n ) by MATRIX_1:25;
A5: ( len M1 = n & width M1 = n ) by MATRIX_1:25;
M2 = (1. (K,n)) * M2 by MATRIX_3:20
.= ((M1 ~) * M1) * M2 by A3, Def2
.= (M1 ~) * M1 by A1, A5, A4, MATRIX_3:35
.= 1. (K,n) by A3, Def2 ;
hence M1 commutes_with M2 by Th6; :: thesis: verum