Addition of quadratic form in linear algebra

Question

I want to prove or disprove that $x^T (A+B) x = x^T Ax + x^T Bx$ where $x \in \mathbb{R}^l$ and $A,B$ are $l\times l$ matrices. I think this statement is true, but the only way I can think of as proving is writing out the matrices vectors and doing algebra. Is this the only way? Can you guys think of more clever way to prove this statement?

Answer 1

Sometimes some proofs use nothing but simple ideas with no "cleverness" involved. With that being said, all you need is the distributive property:

\begin{align*} x^T (A+B)x &= x^T (Ax+Bx) \\ &= x^T Ax + x^T Bx \end{align*}

Done.

Answer 2

Matrix multiplication is distributive with respect to product; i.e. given three matrix $A,B,C$ then $A(B+C)=AB+AC$ and $(A+B)C=AC+BC$. (Of course, when the dimensions make sense to multiply). This being said, you can think of $x^T$ and $x$ as matrix, and using the algebra explained you get $$ x^T(A+B)x=(x^TA+x^TB)x=x^TAx+x^TBx. $$