Find the matrix associated to a given quadratic form

Question

Please help me complete my answer and find the matrix associated to a given quadratic form, here is my work and effort on that

Answer 1

HINT

Note that for the 2-by-2 case

$$v^TAv=\begin{bmatrix}a&b\end{bmatrix}\begin{bmatrix}1&0\\0&1\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix}=a^2+b^2$$