Question

Evaluate the determinant of the matrix: \[ \left| \begin{array}{cc} 1 & \tan x
-\tan x & 1 \end{array} \right| \]

• A. \( 1 - \tan^2 x \)
• B. \( 1 + \tan^2 x \)
• C. \( \sec^2 x \)
• D. \( 0 \)

Hint:

For a 2x2 matrix \( \left| \begin{array}{cc} a & b
c & d \end{array} \right| \), use the formula \( ad - bc \). Pay attention to the signs, especially when trigonometric functions are involved.

Answer 1

The Correct Option is A

The determinant of a \(2 \times 2\) matrix is calculated as follows: \[ \left| \begin{array}{cc} a & b
c & d \end{array} \right| = ad - bc \] Given \( a = 1, b = \tan x, c = -\tan x, d = 1 \), the determinant is computed as: \[ \text{Determinant} = (1)(1) - (\tan x)(-\tan x) = 1 + \tan^2 x \] Therefore, the correct answer is: Correct Answer (Updated): (B) \( 1 + \tan^2 x \)