MathAlgebraLinear algebraLinear operators

Linear operators and matrices

Double rotation

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The problem might be tricky. To solve it, you need to be familiar with trigonometric functions.

Let $\mathcal{A_{\varphi}}$ be rotation operator that rotates the plane by an angle $\varphi$ . Here's the matrix of $\mathcal{A_{\varphi}}$ in the standard basis, $e_1 = (1, 0)$ , $e_2 = (0, 1)$ :
$A = \begin{pmatrix} \cos \varphi & - \sin\varphi \\ \sin \varphi & \cos \varphi \end{pmatrix}.$

What is the matrix of the operator $\mathcal{A}_\varphi\mathcal{A}_{\psi}$ (the operator that applies $\mathcal{A}_\psi$ first and $\mathcal{A}_\varphi$ second)?

Select one option from the list

\begin{pmatrix} \cos \varphi& - \sin \psi \\ \sin \psi & \cos \varphi \end{pmatrix}

\begin{pmatrix} \cos (\varphi * \psi)& - \sin(\varphi * \psi) \\ \sin (\varphi * \psi) & \cos (\varphi * \psi) \end{pmatrix}

\begin{pmatrix} \cos (\varphi + \psi)& - \sin(\varphi + \psi) \\ \sin (\varphi + \psi) & \cos (\varphi + \psi) \end{pmatrix}

\begin{pmatrix} \cos (\varphi - \psi)& - \sin(\varphi - \psi) \\ \sin (\varphi - \psi) & \cos (\varphi - \psi) \end{pmatrix}

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