let n be Nat; :: thesis: 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; :: thesis: 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; :: 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;

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; :: thesis: verum

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; :: thesis: 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; :: 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;

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; :: thesis: verum