let n be Nat; for K being Field
for M1, M2 being Matrix of n,K st M2 is invertible & M1 is_similar_to M2 holds
M1 is invertible
let K be Field; for M1, M2 being Matrix of n,K st M2 is invertible & M1 is_similar_to M2 holds
M1 is invertible
let M1, M2 be Matrix of n,K; ( 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
; M1 is invertible
consider M4 being Matrix of n,K such that
A3:
M4 is invertible
and
A4:
M1 = ((M4 ~) * M2) * M4
by A2;
M4 ~ is invertible
by A3;
then
(M4 ~) * M2 is invertible
by A1, MATRIX_6:36;
hence
M1 is invertible
by A3, A4, MATRIX_6:36; verum