If for the matrix A= [ cc x & 1 -1 & x ], A+A^ =2 3, , then the v… - Vidyadip

MCQ

If for the matrix $A=\left[\begin{array}{cc}\tan x & 1 \\ -1 & \tan x\end{array}\right], A+A^{\prime}=2 \sqrt{3}$, , then the value of $x \in\left[0, \frac{\pi}{2}\right]$ is :

A
0
B
$\frac{\pi}{4}$
C
$\frac{\pi}{3}$
D
$\frac{\pi}{6}$

✓

Answer

We have, $A=\left[\begin{array}{cc}\tan x & 1 \\ -1 & \tan x\end{array}\right]$ $A+A^{\prime}=2 \sqrt{3}$ I
$\Rightarrow\left[\begin{array}{cc}\tan x & 1 \\ -1 & \tan x\end{array}\right]+\left[\begin{array}{cc}\tan x & -1 \\ 1 & \tan x\end{array}\right]=2 \sqrt{3}\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$\Rightarrow\left[\begin{array}{cc}2 \tan x & 0 \\ 0 & 2 \tan x\end{array}\right]=\left[\begin{array}{cc}2 \sqrt{3} & 0 \\ 0 & 2 \sqrt{3}\end{array}\right]$
On comparing, we get
\[\begin{array}{l}
2 \tan x=2 \sqrt{3} \Rightarrow \tan x=\sqrt{3}=\tan \frac{\pi}{3} \\
\Rightarrow \quad x=\frac{\pi}{3}
\end{array}\]

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