Let $A$ be an $m \times n$ matrix and let $B, C$ be $n \times p$ matrices. Prove that $A(B + C) = AB + AC$

Question

Let $A$ be an $m \times n$ matrix and let $B, C$ be $n \times p$ matrices. Prove that $A(B + C) = AB + AC$

I know it's obvious that it is and that every mathematician takes this for granted but I've been asked to prove it and I don't know how to do it without just multiplying out the brackets. Any help would be greatly appreciated.

Answer 1

The $(i, j)$th entry of the left hand side is $$ \sum_{k = 1}^n a_{ik}(b_{kj} + c_{kj})$$ while the $(i, j)$th entry of the right hand side is $$ \sum_{k = 1}^n a_{ik}b_{kj} + \sum_{k = 1}^n a_{ik} c_{kj}$$ which are indeed equal. And since every entry is equal, the matrices must be equal.

Answer 2

Well, what I would do would be tu compute $A(B+C)$ and $AB+AC$ and to see that the result is the same.

Otherwise, you can use the fact that $A$, $B$, and $C$ are matrices of linear maps $f_A$, $f_B$, and $f_C$, that$$f_C\circ(f_B+f_C)=f_A\circ f_C+f_A\circ f_C$$and that the matrix of the composition of two linear maps is the product of the matrices.