Bilinear form with parameter

Question

I have a bilinear form such that the associate matrix is $\ A=\left(\begin{matrix}0&0&k\\ 0&k&0\\ k&0&0\\ \end{matrix}\right)$. $\,$ Does exist a $k$ such that $F_k((e_1+e_2+e_3),(e_1+e_2+e_3))=4$

I don't understand how can I start!

Answer 1

You have that $F_k(\vec{x},\vec{y}) = \vec{x}^\top A \vec{y}$, so $$\begin{align} F_k((1,1,1),(1,1,1)) &= \begin{pmatrix} 1 & 1 & 1\end{pmatrix} \begin{pmatrix} 0 & 0 & k \\ 0 & k & 0 \\ k & 0 & 0\end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \\ &= \begin{pmatrix} 1 & 1 & 1\end{pmatrix} \begin{pmatrix} k \\ k \\ k \end{pmatrix} \\ &= 3k\end{align}$$So $k = 4/3$ fits the bill.