Question

Assertion (A): For the matrix \[ A = \begin{bmatrix} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{bmatrix}, \quad \text{where } \theta \in [0, 2\pi], \] \(|A| \in [2, 4].\)

Reason (R): \(\cos \theta \in [-1, 1], \ \forall \ \theta \in [0, 2\pi].\)

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is \textit{not} the correct explanation of Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

When calculating determinants, simplify using cofactor expansion and evaluate the minors carefully. Use the properties of trigonometric functions like \(\cos \theta \in [-1, 1]\) to determine the range of values for expressions involving \(\cos^2 \theta\).

Answer 1

The Correct Option is A

Step 1: Compute the determinant of matrix \( A \).

\[ |A| = \begin{vmatrix} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{vmatrix}. \]

Using cofactor expansion along the first row:

\[ |A| = 1 \cdot \begin{vmatrix} 1 & \cos \theta \\ -\cos \theta & 1 \end{vmatrix} - \cos \theta \cdot \begin{vmatrix} -\cos \theta & \cos \theta \\ -1 & 1 \end{vmatrix} + 1 \cdot \begin{vmatrix} -\cos \theta & 1 \\ -1 & -\cos \theta \end{vmatrix}. \]

1. Compute the first minor:

\[ \begin{vmatrix} 1 & \cos \theta \\ -\cos \theta & 1 \end{vmatrix} = (1)(1) - (-\cos \theta)(\cos \theta) = 1 + \cos^2 \theta. \]

2. Compute the second minor:

\[ \begin{vmatrix} -\cos \theta & \cos \theta \\ -1 & 1 \end{vmatrix} = (-\cos \theta)(1) - (\cos \theta)(-1) = -\cos \theta + \cos \theta = 0. \]

3. Compute the third minor:

\[ \begin{vmatrix} -\cos \theta & 1 \\ -1 & -\cos \theta \end{vmatrix} = (-\cos \theta)(-\cos \theta) - (1)(-1) = \cos^2 \theta + 1. \]

Substitute back into the determinant:

\[ |A| = 1 \cdot (1 + \cos^2 \theta) - \cos \theta \cdot 0 + 1 \cdot (1 + \cos^2 \theta). \]

Simplify:

\[ |A| = (1 + \cos^2 \theta) + (1 + \cos^2 \theta) = 2 + 2\cos^2 \theta. \]

Step 2: Determine the range of \(|A| \).

Since \(\cos \theta \in [-1, 1]\), we have:

\[ \cos^2 \theta \in [0, 1]. \]

Thus:

\[ |A| = 2 + 2\cos^2 \theta \in [2, 4]. \]

Verification of Assertion (A): The determinant \(|A|\) lies in the interval \([2, 4]\), so the assertion is true.

Verification of Reason (R): The cosine function satisfies \(\cos \theta \in [-1, 1]\) for all \(\theta \in [0, 2\pi]\), so the reason is also true.

Conclusion: Both Assertion (A) and Reason (R) are true, and the Reason (R) correctly explains the Assertion (A).