let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 is Nilpotent & M1 commutes_with M2 & n > 0 holds
M1 * M2 is Nilpotent
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is Nilpotent & M1 commutes_with M2 & n > 0 holds
M1 * M2 is Nilpotent
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is Nilpotent & M1 commutes_with M2 & n > 0 implies M1 * M2 is Nilpotent )
assume A1: ( M1 is Nilpotent & M1 commutes_with M2 & n > 0 ) ; :: thesis: M1 * M2 is Nilpotent
A2: ( len M1 = n & width M1 = n & len M2 = n & width M2 = n ) by MATRIX_1:25;
A3: ( len (M2 * M1) = n & width (M2 * M1) = n ) by MATRIX_1:25;
(M1 * M2) * (M1 * M2) = (M2 * M1) * (M1 * M2) by A1, MATRIX_6:def 1
.= ((M2 * M1) * M1) * M2 by A2, A3, MATRIX_3:35
.= (M2 * (M1 * M1)) * M2 by A2, MATRIX_3:35
.= (M2 * (0. K,n)) * M2 by A1, Def2
.= (0. K,n,n) * M2 by A1, A2, MATRIX_6:2
.= 0. K,n by A2, MATRIX_6:1 ;
hence M1 * M2 is Nilpotent by Def2; :: thesis: verum