let n be Nat; :: thesis: for K being Field

for M1, M2 being Matrix of n,K st M1 is invertible & M1 commutes_with M2 holds

M1 ~ commutes_with M2

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is invertible & M1 commutes_with M2 holds

M1 ~ commutes_with M2

let M1, M2 be Matrix of n,K; :: thesis: ( M1 is invertible & M1 commutes_with M2 implies M1 ~ commutes_with M2 )

assume that

A1: M1 is invertible and

A2: M1 commutes_with M2 ; :: thesis: M1 ~ commutes_with M2

A3: M1 ~ is_reverse_of M1 by A1, Def4;

A4: width M2 = n by MATRIX_0:24;

A5: ( width M1 = n & len M1 = n ) by MATRIX_0:24;

A6: ( len (M2 * M1) = n & width (M2 * M1) = n ) by MATRIX_0:24;

A7: len (M1 ~) = n by MATRIX_0:24;

A8: len M2 = n by MATRIX_0:24;

A9: width (M1 ~) = n by MATRIX_0:24;

M2 = (1. (K,n)) * M2 by MATRIX_3:18

.= ((M1 ~) * M1) * M2 by A3

.= (M1 ~) * (M1 * M2) by A5, A8, A9, MATRIX_3:33

.= (M1 ~) * (M2 * M1) by A2 ;

then M2 * (M1 ~) = (M1 ~) * ((M2 * M1) * (M1 ~)) by A9, A7, A6, MATRIX_3:33

.= (M1 ~) * (M2 * (M1 * (M1 ~))) by A5, A4, A7, MATRIX_3:33

.= (M1 ~) * (M2 * (1. (K,n))) by A3

.= (M1 ~) * M2 by MATRIX_3:19 ;

hence M1 ~ commutes_with M2 ; :: thesis: verum

for M1, M2 being Matrix of n,K st M1 is invertible & M1 commutes_with M2 holds

M1 ~ commutes_with M2

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is invertible & M1 commutes_with M2 holds

M1 ~ commutes_with M2

let M1, M2 be Matrix of n,K; :: thesis: ( M1 is invertible & M1 commutes_with M2 implies M1 ~ commutes_with M2 )

assume that

A1: M1 is invertible and

A2: M1 commutes_with M2 ; :: thesis: M1 ~ commutes_with M2

A3: M1 ~ is_reverse_of M1 by A1, Def4;

A4: width M2 = n by MATRIX_0:24;

A5: ( width M1 = n & len M1 = n ) by MATRIX_0:24;

A6: ( len (M2 * M1) = n & width (M2 * M1) = n ) by MATRIX_0:24;

A7: len (M1 ~) = n by MATRIX_0:24;

A8: len M2 = n by MATRIX_0:24;

A9: width (M1 ~) = n by MATRIX_0:24;

M2 = (1. (K,n)) * M2 by MATRIX_3:18

.= ((M1 ~) * M1) * M2 by A3

.= (M1 ~) * (M1 * M2) by A5, A8, A9, MATRIX_3:33

.= (M1 ~) * (M2 * M1) by A2 ;

then M2 * (M1 ~) = (M1 ~) * ((M2 * M1) * (M1 ~)) by A9, A7, A6, MATRIX_3:33

.= (M1 ~) * (M2 * (M1 * (M1 ~))) by A5, A4, A7, MATRIX_3:33

.= (M1 ~) * (M2 * (1. (K,n))) by A3

.= (M1 ~) * M2 by MATRIX_3:19 ;

hence M1 ~ commutes_with M2 ; :: thesis: verum