Question

Let
$p\left(\theta \right)=\left|\begin{array}{ccc}-\sqrt{2}& \sin \theta & \cos \theta \\ 1& \cos \theta & \sin \theta \\ -1& \sin \theta & -\cos \theta \end{array}\right|$ $q\left(\theta \right)=\sqrt{2}\left|\begin{array}{ccc}\sin 2\theta & 1& 1\\ \cos 2\theta & 2& 3\\ \cos 2\theta & 3& 5\end{array}\right|$ $r\left(\theta \right)=\left|\begin{array}{ccc}\cos \theta & \sin \theta & \cos \theta \\ -\sin \theta & \cos \theta & \sin \theta \\ -\cos \theta & -\sin \theta & \cos \theta \end{array}\right|$ and $s\left(\theta \right)=\left|\begin{array}{ccc}{\sec }^2\theta & 1& 1\\ {\cos }^2\theta & {\cos }^2\theta & {\csc }^2\theta \\ 1& {\cos }^2\theta & {\cot }^2\theta \end{array}\right|$
Match the functions on the left with their range on the right

Answer 1

A) $p\left(\theta \right)=\left|\begin{array}{ccc}-\sqrt{2}& \sin \theta & \cos \theta \\ 1& \cos \theta & \sin \theta \\ -1& \sin \theta & -\cos \theta \end{array}\right|$
Expanding along $C_1$
$p\left(\theta \right)=(-\sqrt{2})(-{\cos }^2\theta -{\sin }^2\theta )-1(-\sin \theta \cos \theta -\sin \theta \cos \theta )-1({\sin }^2\theta -{\cos }^2\theta )=\sqrt{2}+\sin 2\theta +\cos 2\theta =\sqrt{2}+\sqrt{2}\sin (2\theta +\frac{\pi }{4})\Rightarrow p\left(\theta \right)\in [0,2\sqrt{2}]$
B) $q\left(\theta \right)=\sqrt{2}\left|\begin{array}{ccc}\sin 2\theta & 1& 1\\ \cos 2\theta & 2& 3\\ \cos 2\theta & 3& 5\end{array}\right|$
Expanding along $C_1$
$q\left(\theta \right)=\sqrt{2}\left|\begin{array}{ccc}\sin 2\theta & 1& 1\\ \cos 2\theta & 2& 3\\ \cos 2\theta & 3& 5\end{array}\right|=\sqrt{2}\left(\sin 2\theta (10-9)-\cos 2\theta (5-3)+\cos 2\theta (3-2)\right)=\sqrt{2}\left(\sin 2\theta -\cos 2\theta \right)=2\sin (2\theta -\frac{\pi }{2})\Rightarrow q\left(\theta \right)\in [-2,2]$
C) $r\left(\theta \right)=\left|\begin{array}{ccc}\cos \theta & \sin \theta & \cos \theta \\ -\sin \theta & \cos \theta & \sin \theta \\ -\cos \theta & -\sin \theta & \cos \theta \end{array}\right|$
Applying $C_1\to C_1+C_3$
$r\left(\theta \right)=\left|\begin{array}{ccc}2\cos \theta & \sin \theta & \cos \theta \\ 0& \cos \theta & \sin \theta \\ 0& -\sin \theta & \cos \theta \end{array}\right|=2\cos \theta \left|\begin{array}{ccc}1& \sin \theta & \cos \theta \\ 0& \cos \theta & \sin \theta \\ 0& -\sin \theta & \cos \theta \end{array}\right|=2\cos \theta \left(1({\cos }^2\theta +{\sin }^2\theta )\right)=2\cos \theta$
$\Rightarrow r\left(\theta \right)\in [-2,2]$
D) $s\left(\theta \right)=\left|\begin{array}{ccc}{\sec }^2\theta & 1& 1\\ {\cos }^2\theta & {\cos }^2\theta & {\csc }^2\theta \\ 1& {\cos }^2\theta & {\cot }^2\theta \end{array}\right|={\sec }^2\theta \left|\begin{array}{ccc}1& {\cos }^2\theta & {\cos }^2\theta \\ {\cos }^2\theta & {\cos }^2\theta & {\csc }^2\theta \\ 1& {\cos }^2\theta & {\cot }^2\theta \end{array}\right|$
Applying $R_3\to R_3-R_1$
$s\left(\theta \right)={\sec }^2\theta \left|\begin{array}{ccc}1& {\cos }^2\theta & {\cos }^2\theta \\ {\cos }^2\theta & {\cos }^2\theta & {\csc }^2\theta \\ 0& 0& {\cot }^2\theta -{\cos }^2\theta \end{array}\right|={\sec }^2\theta ({\cot }^2\theta -{\cos }^2\theta )({\cos }^2\theta -{\cos }^4\theta )=({\cot }^2\theta -{\cos }^2\theta ){\sin }^2\theta ={\cos }^4\theta \Rightarrow s\left(\theta \right)\in [0,1]$