I have solved for options 1, 2 and 3 but I'm stuck at the 4th one. Can anyone help?
2026-03-27 14:45:18.1774622718
Can you help me with the 4th option?
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If$$P=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix},$$then$$\begin{bmatrix}x&y&z\end{bmatrix}.P.\begin{bmatrix}x&y&z\end{bmatrix}^T=\begin{bmatrix}ax^2+ey^2+iz^2+(b+d)xy+(c+g)xz+(f+h)yz\end{bmatrix}.\tag1$$On the other hand,$$Q=\begin{bmatrix}a&\frac12(b+d)&\frac12(c+g)\\\frac12(b+d)&e&\frac12(f+h)\\\frac12(c+g)&\frac12(f+h)&i\end{bmatrix}$$and it is clear that if you put $Q$ instead of $P$ in $(1)$, that equality still holds.