To prove that $(A + B) - (A + C) = B - C$ for matrices $A$, $B$, and $C$ of the same size, we can follow these steps:

1. Start with the left-hand side: $(A + B) - (A + C)$

2. Apply the property of matrix subtraction: According to the property of subtraction, we can rewrite this expression as: $(A + B) + (-1)(A + C)$

3. Distribute the negative sign: This gives us: $(A + B) - A - C$

4. Combine like terms: The terms $A$ and $-A$ cancel each other out, resulting in: $B - C$

5. Conclude: Thus, we have shown that: $(A + B) - (A + C) = B - C$

Therefore, the statement is proven to be true.

If you need further details or have any questions, feel free to ask! Here are some related questions you might consider:

1. What are the properties of matrix addition and subtraction?
2. How does this proof extend to scalar multiplication with matrices?
3. Can you provide examples with specific matrices?
4. What is the significance of matrix dimensions in these operations?
5. How do these properties apply in the context of linear transformations?

Tip: Always remember that matrix operations follow specific rules, and understanding these can help simplify complex proofs and problems.