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

for M1, M2 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 M1, M2 being Matrix of n,K st M2 is invertible & M1 is_congruent_Matrix_of M2 & n > 0 holds

M1 is invertible

let M1, M2 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;

set M6 = (M4 ~) @ ;

set M5 = M4 @ ;

A6: ( width M4 = n & width (M4 ~) = n ) by MATRIX_0:24;

len M4 = n by MATRIX_0:24;

then A7: ((M4 ~) * M4) @ = (M4 @) * ((M4 ~) @) by A3, A6, MATRIX_3:22;

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_0:24;

then (M4 * (M4 ~)) @ = ((M4 ~) @) * (M4 @) by A3, A6, MATRIX_3:22;

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_6:36;

hence M1 is invertible by A4, A5, MATRIX_6:36; :: thesis: verum

for M1, M2 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 M1, M2 being Matrix of n,K st M2 is invertible & M1 is_congruent_Matrix_of M2 & n > 0 holds

M1 is invertible

let M1, M2 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;

set M6 = (M4 ~) @ ;

set M5 = M4 @ ;

A6: ( width M4 = n & width (M4 ~) = n ) by MATRIX_0:24;

len M4 = n by MATRIX_0:24;

then A7: ((M4 ~) * M4) @ = (M4 @) * ((M4 ~) @) by A3, A6, MATRIX_3:22;

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_0:24;

then (M4 * (M4 ~)) @ = ((M4 ~) @) * (M4 @) by A3, A6, MATRIX_3:22;

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_6:36;

hence M1 is invertible by A4, A5, MATRIX_6:36; :: thesis: verum