let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 is Nilpotent & M2 is Nilpotent & M1 * M2 = - (M2 * M1) holds
M1 + M2 is Nilpotent
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is Nilpotent & M2 is Nilpotent & M1 * M2 = - (M2 * M1) holds
M1 + M2 is Nilpotent
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is Nilpotent & M2 is Nilpotent & M1 * M2 = - (M2 * M1) implies M1 + M2 is Nilpotent )
assume that
A1: ( M1 is Nilpotent & M2 is Nilpotent ) and
A2: M1 * M2 = - (M2 * M1) ; :: thesis: M1 + M2 is Nilpotent
A4: ( M1 * M1 = 0. (K,n) & M2 * M2 = 0. (K,n) ) by A1;
A5: ( len (M1 * M1) = n & width (M1 * M1) = n ) by MATRIX_0:24;
A6: ( len M1 = n & width M1 = n ) by MATRIX_0:24;
A7: ( len ((M1 * M1) + (M2 * M1)) = n & width ((M1 * M1) + (M2 * M1)) = n ) by MATRIX_0:24;
A8: ( len (M2 * M1) = n & width (M2 * M1) = n ) by MATRIX_0:24;
A9: ( len (M1 * M2) = n & width (M1 * M2) = n ) by MATRIX_0:24;
A10: ( len M2 = n & width M2 = n ) by MATRIX_0:24;
( len (M1 + M2) = n & width (M1 + M2) = n ) by MATRIX_0:24;
then (M1 + M2) * (M1 + M2) = ((M1 + M2) * M1) + ((M1 + M2) * M2) by A6, A10, MATRIX_4:62
.= ((M1 * M1) + (M2 * M1)) + ((M1 + M2) * M2) by A6, A10, MATRIX_4:63
.= ((M1 * M1) + (M2 * M1)) + ((M1 * M2) + (M2 * M2)) by A6, A10, MATRIX_4:63
.= (((M1 * M1) + (M2 * M1)) + (M1 * M2)) + (M2 * M2) by A9, A7, MATRIX_3:3
.= ((M1 * M1) + ((M2 * M1) + (- (M2 * M1)))) + (M2 * M2) by A2, A5, A8, MATRIX_3:3
.= ((M1 * M1) + (0. (K,n,n))) + (M2 * M2) by A8, MATRIX_4:2
.= (0. (K,n)) + (0. (K,n)) by A4, MATRIX_3:4
.= 0. (K,n,n) by MATRIX_3:4
.= 0. (K,n) ;
hence M1 + M2 is Nilpotent ; :: thesis: verum