let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 is invertible & M2 * M1 = 0. K,n holds
M2 = 0. K,n
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is invertible & M2 * M1 = 0. K,n holds
M2 = 0. K,n
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is invertible & M2 * M1 = 0. K,n implies M2 = 0. K,n )
assume that
A1: M1 is invertible and
A2: M2 * M1 = 0. K,n ; :: thesis: M2 = 0. K,n
A3: M1 ~ is_reverse_of M1 by A1, MATRIX_6:def 4;
A4: width M2 = n by MATRIX_1:25;
A5: ( width M1 = n & len M1 = n ) by MATRIX_1:25;
A6: width (M1 ~ ) = n by MATRIX_1:25;
A7: len (M1 ~ ) = n by MATRIX_1:25;
M2 = M2 * (1. K,n) by MATRIX_3:21
.= M2 * (M1 * (M1 ~ )) by A3, MATRIX_6:def 2
.= (M2 * M1) * (M1 ~ ) by A5, A4, A7, MATRIX_3:35
.= 0. K,n,n by A2, A6, A7, MATRIX_6:1
.= 0. K,n ;
hence M2 = 0. K,n ; :: thesis: verum