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

for M1, M2 being Matrix of n,K st n > 0 & M1 commutes_with M2 holds

M1 + M1 commutes_with M2

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st n > 0 & M1 commutes_with M2 holds

M1 + M1 commutes_with M2

let M1, M2 be Matrix of n,K; :: thesis: ( n > 0 & M1 commutes_with M2 implies M1 + M1 commutes_with M2 )

assume that

A1: n > 0 and

A2: M1 commutes_with M2 ; :: thesis: M1 + M1 commutes_with M2

A3: width M2 = n by MATRIX_0:24;

A4: len M1 = n by MATRIX_0:24;

A5: ( width M1 = n & len M2 = n ) by MATRIX_0:24;

then (M1 + M1) * M2 = (M1 * M2) + (M1 * M2) by A1, A4, MATRIX_4:63

.= (M2 * M1) + (M1 * M2) by A2

.= (M2 * M1) + (M2 * M1) by A2

.= M2 * (M1 + M1) by A1, A5, A3, A4, MATRIX_4:62 ;

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

for M1, M2 being Matrix of n,K st n > 0 & M1 commutes_with M2 holds

M1 + M1 commutes_with M2

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st n > 0 & M1 commutes_with M2 holds

M1 + M1 commutes_with M2

let M1, M2 be Matrix of n,K; :: thesis: ( n > 0 & M1 commutes_with M2 implies M1 + M1 commutes_with M2 )

assume that

A1: n > 0 and

A2: M1 commutes_with M2 ; :: thesis: M1 + M1 commutes_with M2

A3: width M2 = n by MATRIX_0:24;

A4: len M1 = n by MATRIX_0:24;

A5: ( width M1 = n & len M2 = n ) by MATRIX_0:24;

then (M1 + M1) * M2 = (M1 * M2) + (M1 * M2) by A1, A4, MATRIX_4:63

.= (M2 * M1) + (M1 * M2) by A2

.= (M2 * M1) + (M2 * M1) by A2

.= M2 * (M1 + M1) by A1, A5, A3, A4, MATRIX_4:62 ;

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