let n be Nat; :: thesis: for K being Field
for M1 being Matrix of n,K st M1 is Nilpotent & n > 0 holds
M1 @ is Nilpotent
let K be Field; :: thesis: for M1 being Matrix of n,K st M1 is Nilpotent & n > 0 holds
M1 @ is Nilpotent
let M1 be Matrix of n,K; :: thesis: ( M1 is Nilpotent & n > 0 implies M1 @ is Nilpotent )
assume A1: ( M1 is Nilpotent & n > 0 ) ; :: thesis: M1 @ is Nilpotent
( len M1 = n & width M1 = n ) by MATRIX_1:25;
then (M1 @ ) * (M1 @ ) = (M1 * M1) @ by A1, MATRIX_3:24
.= (0. K,n) @ by A1, Def2
.= n,n --> (0. K) by MATRIX_6:21
.= 0. K,n ;
hence M1 @ is Nilpotent by Def2; :: thesis: verum