Question

prove this law of matrix algebra :-
A(B+C)=AB+AC

Answer 1

should all be n*n matrices ?

Answer 2

or its ok ?

Answer 3

i proved it for only nxn though , does that work ?

Answer 4

to me, B, C must be the same size, but A can be not
I mean, if the size of B, C is n x n, and the size of A is m x n, the expression hods, still

Answer 5

*holds

Answer 6

look at the (i,j) entry of A(B+C)

the ij entry of B+C is \(b_{ij}+c_{ij}\)
the ij entry of A(B+C) is \(\displaystyle\sum_{k=1}^{n}a_{ik}(b_{kj}+c_{kj})\)

then ij entry of AB is \(\displaystyle\sum_{k=1}^{n}a_{ik}b_{kj}\)

then ij entry of AC is \(\displaystyle\sum_{k=1}^{n}a_{ik}c_{kj}\)

and so the ij entry of AB+AC is


\(\displaystyle\sum_{k=1}^{n}a_{ik}b_{kj}+\sum_{k=1}^{n}a_{ik}c_{kj}\)

\(=\displaystyle\sum_{k=1}^{n}a_{ik}(b_{kj}+c_{kj})\)

Answer 7

okk thank you soo much ^_^