<s>[INST] how to prove this question of eigen-values and eigen-vectors? If
$$
A=\begin{bmatrix}
\sin\theta & \csc\theta & 1 \\
\sec\theta & \cos\theta & 1 \\
\tan\theta & \cot\theta & 1 \\
\end{bmatrix}
$$
then prove that there does not exist a real value of $\theta$ for which characteristics roots of $A$ are $-1,1,3$
i tried to solve as follows, 
sum of eigen value $$=\sin\theta +\cos\theta + 1=-1+1+3=3$$
$$\sin\theta +\cos\theta = 2$$
but what to do next.
 [/INST] \begin{eqnarray}
a\sin\theta+b\cos\theta&=&\sqrt{a^2+b^2}\left[\frac{a}{\sqrt{a^2+b^2}}\sin\theta+\frac{b}{\sqrt{a^2+b^2}}\cos\theta\right]\\
&=&\sqrt{a^2+b^2}\left[\sin\phi\sin\theta+\cos\phi\cos\theta\right]\\
&=&\sqrt{a^2+b^2}\cos(\theta-\phi)
\end{eqnarray}
Where $$\phi=\arcsin\frac{a}{\sqrt{a^2+b^2}}$$

So the result is a sinusoidal with a phase shift and an amplitude of $\sqrt{a^2+b^2}$
When $a=b=1$ what will be the amplitude?
</s>