Matrices: If A=beginbmatrixcostheta&-sintheta\\sintheta&costhetaendbmatrix ,… | JEE Main 2019

JEE Main 2019 (Online) 9th January Morning ShiftQuestion 1

If $A=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$, then the matrix $A^{-50}$ when $\theta=\frac\pi{12}$, is equal to:

Matrices

$\begin{bmatrix}\frac{\sqrt3}{2}&\frac12\\-\frac12&\frac{\sqrt3}{2}\end{bmatrix}$✓ Correct answer$\begin{bmatrix}\frac12&\frac{\sqrt3}{2}\\-\frac{\sqrt3}{2}&\frac12\end{bmatrix}$$\begin{bmatrix}\frac{\sqrt3}{2}&-\frac12\\\frac12&\frac{\sqrt3}{2}\end{bmatrix}$$\begin{bmatrix}\frac12&-\frac{\sqrt3}{2}\\\frac{\sqrt3}{2}&\frac12\end{bmatrix}$

Solution

$A^{-50}$ is rotation by $-50\cdot\frac\pi{12}=-\frac{25\pi}6\equiv-\frac\pi6$, i.e. $\begin{bmatrix}\cos\frac\pi6&\sin\frac\pi6\\-\sin\frac\pi6&\cos\frac\pi6\end{bmatrix}$.

Matrices - JEE Mathematics question (JEE Main 2019 (Online) 9th January Morning Shift)

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