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

for M1, M2 being Matrix of n,K st M1 commutes_with M2 holds

M1 * M2 commutes_with M2

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 commutes_with M2 holds

M1 * M2 commutes_with M2

let M1, M2 be Matrix of n,K; :: thesis: ( M1 commutes_with M2 implies M1 * M2 commutes_with M2 )

A1: ( width M1 = n & width M2 = n ) by MATRIX_0:24;

A2: ( len M1 = n & len M2 = n ) by MATRIX_0:24;

assume M1 commutes_with M2 ; :: thesis: M1 * M2 commutes_with M2

then (M1 * M2) * M2 = M2 * (M1 * M2) by A1, A2, MATRIX_3:33;

hence M1 * M2 commutes_with M2 ; :: thesis: verum

for M1, M2 being Matrix of n,K st M1 commutes_with M2 holds

M1 * M2 commutes_with M2

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 commutes_with M2 holds

M1 * M2 commutes_with M2

let M1, M2 be Matrix of n,K; :: thesis: ( M1 commutes_with M2 implies M1 * M2 commutes_with M2 )

A1: ( width M1 = n & width M2 = n ) by MATRIX_0:24;

A2: ( len M1 = n & len M2 = n ) by MATRIX_0:24;

assume M1 commutes_with M2 ; :: thesis: M1 * M2 commutes_with M2

then (M1 * M2) * M2 = M2 * (M1 * M2) by A1, A2, MATRIX_3:33;

hence M1 * M2 commutes_with M2 ; :: thesis: verum