GATE (TF) Textile 2013 Question Paper Solution | GATE/2013/TF/34

Question 34 (Textile Engineering & Fibre Science)

The inverse of the matrix \begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix} is

(A) \begin{bmatrix}\sin\theta & \cos\theta \\ \cos\theta & -\sin\theta\end{bmatrix}
(B) \begin{bmatrix}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta\end{bmatrix}
(C) \begin{bmatrix}-\sin\theta & \cos\theta \\ \cos\theta & \sin\theta\end{bmatrix}
(D) \begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}
Answer / Solution

Given-
A=\begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}
A^{-1}=\frac{Adj A}{\left |A \right |}
\left |A \right |=Cos^2\theta+Sin^2\theta=1
Adj A=\begin{bmatrix}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta\end{bmatrix}
A^{-1}=\frac{\begin{bmatrix}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta\end{bmatrix}}{\left |1 \right |}
A^{-1}=\begin{bmatrix}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta\end{bmatrix} (Ans)

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