Question

Check whether the matrix 

 is invertible or not.

Hint:

A matrix is invertible if and only if its determinant is nonzero.

Answer 1

Step 1: Calculate the Determinant 
For a \( 2 \times 2 \) matrix A =

, the determinant is given by: \[ \det(A) = (a d - b c). \] Substituting the values from matrix \( A \): \[ \det(A) = (\cos\theta \cdot \cos\theta - \sin\theta \cdot (-\sin\theta)) \] \[ = \cos^2\theta + \sin^2\theta. \] Step 2: Check the Invertibility Condition 
Since \( \cos^2\theta + \sin^2\theta = 1 \), we have: \[ \det(A) = 1 \neq 0. \] As the determinant is nonzero, the matrix is invertible.