Question

If \(A = [\begin{array}{cc}secθ & -tanθ \\ -tanθ & secθ\end{array}]\), \(θ\in (0,\frac{π}{2})\) such that \(A+adj\,A = 4I\), then \(θ =\)

• A. \(\frac{π}{12}\)
• B. \(\frac{π}{6}\)
• C. \(\frac{π}{3}\)
• D. \(\frac{π}{4}\)

Hint:

For a 2 by 2 matrix, \(\text{adj}A\) swaps the diagonal and changes signs of the off-diagonal entries.

Answer 1

The Correct Option is C

Step 1: Key Formula:
For \(A = \begin{bmatrix}a & b\\ c & d\end{bmatrix}\), \(\text{adj}\,A = \begin{bmatrix}d & -b\\ -c & a\end{bmatrix}\).

Step 2: Find adj A:
Here \(a = d = \sec\theta\) and \(b = c = -\tan\theta\). So \(\text{adj}\,A = \begin{bmatrix}\sec\theta & \tan\theta\\ \tan\theta & \sec\theta\end{bmatrix}\).

Step 3: Add:
\(A + \text{adj}\,A = \begin{bmatrix}2\sec\theta & 0\\ 0 & 2\sec\theta\end{bmatrix} = 2\sec\theta\, I\).
Set \(2\sec\theta = 4\), so \(\sec\theta = 2\) and \(\cos\theta = \frac12\). With \(\theta\in(0,\frac{\pi}{2})\), \(\theta = \frac{\pi}{3}\).

Final Answer:
The value is \(\theta = \frac{\pi}{3}\), option (C). \[ \boxed{\frac{\pi}{3}} \]