If \(A = \begin{bmatrix}2 & \cos\theta & 1\\ -\cos\theta & 1 & \cos\theta\\ -1 & -\cos\theta & 2\end{bmatrix}\), then for all real \(\theta\), \(|A|\) lies in the interval
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \([5, 9]\)
Let c stand for \(\cos\theta\). Expanding along the first row:
\(|A| = 2\big(1 \cdot 2 - c(-c)\big) - c\big((-c)(2) - c(-1)\big) + 1\big((-c)(-c) - 1 \cdot (-1)\big)\)
\(|A| = 2(2 + c^{2}) - c(-c) + (c^{2} + 1)\)
\(|A| = 4 + 2c^{2} + c^{2} + c^{2} + 1\)
\(|A| = 5 + 4\cos^{2}\theta\)
Since \(0 \le \cos^{2}\theta \le 1\), the least value is 5 (at \(\theta = \frac{\pi}{2}\)) and the greatest value is 9 (at \(\theta = 0\)).
So \(|A|\) lies in \([5, 9]\).