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 validate the assertion and reason, we compute the determinant of matrix \( A \): \[ |A| = \begin{vmatrix} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{vmatrix}. \] By using cofactor expansion along the first row, we get: \[ |A| = 1 \cdot \begin{vmatrix} 1 & \cos \theta \\ -\cos \theta & 1 \end{vmatrix} - \cos \theta \cdot \begin{vmatrix} -\cos \theta & \cos \theta \\ -1 & 1 \end{vmatrix} + 1 \cdot \begin{vmatrix} -\cos \theta & 1 \\ -1 & -\cos \theta \end{vmatrix}. \] First minor calculation: \[ \begin{vmatrix} 1 & \cos \theta \\ -\cos \theta & 1 \end{vmatrix} = (1)(1) - (-\cos \theta)(\cos \theta) = 1 + \cos^2 \theta. \] Second minor calculation: \[ \begin{vmatrix} -\cos \theta & \cos \theta \\ -1 & 1 \end{vmatrix} = (-\cos \theta)(1) - (\cos \theta)(-1) = -\cos \theta + \cos \theta = 0. \] Third minor calculation: \[ \begin{vmatrix} -\cos \theta & 1 \\ -1 & -\cos \theta \end{vmatrix} = (-\cos \theta)(-\cos \theta) - (1)(-1) = \cos^2 \theta + 1. \] Substituting these minors back into the determinant equation: \[ |A| = 1 \cdot (1 + \cos^2 \theta) - \cos \theta \cdot 0 + 1 \cdot (1 + \cos^2 \theta). \] Simplifying the expression: \[ |A| = (1 + \cos^2 \theta) + (1 + \cos^2 \theta) = 2 + 2\cos^2 \theta. \] Given that \(\cos \theta \in [-1, 1]\), it follows that \(\cos^2 \theta \in [0, 1]\). Therefore, the determinant \(|A|\) ranges from: \[ |A| = 2 + 2\cos^2 \theta \in [2, 4]. \] Verification of Assertion (A): The determinant \(|A|\) falls within the interval \([2, 4]\), confirming the assertion is true. Verification of Reason (R): The cosine function's range is \(\cos \theta \in [-1, 1]\) for all \(\theta \in [0, 2\pi]\), hence the reason is also true. Conclusion: Both Assertion (A) and Reason (R) are true, and Reason (R) provides a correct explanation for Assertion (A).