let n be Nat; :: thesis: for K being Field
for M2, M1 being Matrix of n,K st M2 is invertible & M1 is_similar_to M2 holds
M1 is invertible
let K be Field; :: thesis: for M2, M1 being Matrix of n,K st M2 is invertible & M1 is_similar_to M2 holds
M1 is invertible
let M2, M1 be Matrix of n,K; :: thesis: ( M2 is invertible & M1 is_similar_to M2 implies M1 is invertible )
assume A1: ( M2 is invertible & M1 is_similar_to M2 ) ; :: thesis: M1 is invertible
then consider M4 being Matrix of n,K such that
A2: ( M4 is invertible & M1 = ((M4 ~ ) * M2) * M4 ) by Def5;
( M4 ~ is invertible & (M4 ~ ) ~ = M4 ) by A2, MATRIX_6:16;
then (M4 ~ ) * M2 is invertible by A1, MATRIX_6:37;
hence M1 is invertible by A2, MATRIX_6:37; :: thesis: verum