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

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

To verify the given assertion and reason, we calculate 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: \[ |A| = (1 + \cos^2 \theta) + (1 + \cos^2 \theta) = 2 + 2\cos^2 \theta. \] 

Since \(\cos \theta \in [-1, 1]\), we have: \[ \cos^2 \theta \in [0, 1]. \] Thus, the determinant \(|A|\) varies as: \[ |A| = 2 + 2\cos^2 \theta \in [2, 4]. \] 

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).