We are given that,
\(A=\begin{vmatrix} cos\,\theta & sin\,\theta \\ -sin\,\theta & cos\,\theta \\ \end{vmatrix}\)
We need to find the det(An).
To find det(An),
First we need to find An, and then take determinant of An.
Let us find A2.
A2 = A.A
For z11: Dot multiply the first row of the first matrix and first column of the second matrix, then sum up
That is,
Now, taking determinant of An,
Determinant of 2 × 2 matrix is found as,
So,
Det(An) = cos nθ × cos nθ – sin nθ × (-sin nθ)
\(\Rightarrow\) Det(An) = cos2 nθ + sin2 nθ
Using the algebraic identity,
sin2 A + cos2 A = 1
⇒ Det(An) = 1
Thus, Det(An) is 1.