let n be Nat; :: thesis: for K being Field
for M2, M1 being Matrix of n,K st M2 is invertible & M1 is_congruent_Matrix_of M2 & n > 0 holds
M1 is invertible
let K be Field; :: thesis: for M2, M1 being Matrix of n,K st M2 is invertible & M1 is_congruent_Matrix_of M2 & n > 0 holds
M1 is invertible
let M2, M1 be Matrix of n,K; :: thesis: ( M2 is invertible & M1 is_congruent_Matrix_of M2 & n > 0 implies M1 is invertible )
assume that
A1: M2 is invertible and
A2: M1 is_congruent_Matrix_of M2 and
A3: n > 0 ; :: thesis: M1 is invertible
consider M4 being Matrix of n,K such that
A4: M4 is invertible and
A5: M1 = ((M4 @ ) * M2) * M4 by A2, Def6;
set M6 = (M4 ~ ) @ ;
set M5 = M4 @ ;
A6: ( width M4 = n & width (M4 ~ ) = n ) by MATRIX_1:25;
len M4 = n by MATRIX_1:25;
then A7: ((M4 ~ ) * M4) @ = (M4 @ ) * ((M4 ~ ) @ ) by A3, A6, MATRIX_3:24;
A8: M4 ~ is_reverse_of M4 by A4, MATRIX_6:def 4;
then (M4 ~ ) * M4 = 1. K,n by MATRIX_6:def 2;
then A9: (M4 @ ) * ((M4 ~ ) @ ) = 1. K,n by A7, MATRIX_6:10;
len (M4 ~ ) = n by MATRIX_1:25;
then (M4 * (M4 ~ )) @ = ((M4 ~ ) @ ) * (M4 @ ) by A3, A6, MATRIX_3:24;
then (M4 @ ) * ((M4 ~ ) @ ) = ((M4 ~ ) @ ) * (M4 @ ) by A8, A7, MATRIX_6:def 2;
then M4 @ is_reverse_of (M4 ~ ) @ by A9, MATRIX_6:def 2;
then M4 @ is invertible by MATRIX_6:def 3;
then (M4 @ ) * M2 is invertible by A1, MATRIX_6:37;
hence M1 is invertible by A4, A5, MATRIX_6:37; :: thesis: verum