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 that
A1: M2 is invertible and
A2: M1 is_similar_to M2 ; :: thesis: M1 is invertible
consider M4 being Matrix of n,K such that
A3: M4 is invertible and
A4: M1 = ((M4 ~) * M2) * M4 by A2, Def5;
M4 ~ is invertible by A3, MATRIX_6:16;
then (M4 ~) * M2 is invertible by A1, MATRIX_6:37;
hence M1 is invertible by A3, A4, MATRIX_6:37; :: thesis: verum