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If \(A=\begin{vmatrix} cos\,\theta & sin\,\theta \\ -sin\,\theta & cos\,\theta \\ \end{vmatrix},\) then for any natural number, find the value of Det(An).

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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

Using the algebraic identity,

sin2 A + cos2 A = 1

⇒ Det(An) = 1

Thus, Det(An) is 1.

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