If A=12pmatrix1&√3-√3&1pmatrix, then:

Question

If $A=\dfrac12\begin{pmatrix}1&\sqrt3\-\sqrt3&1\end{pmatrix}$, then:

Accepted Answer

$A=\begin{pmatrix}\cos60^\circ & \sin60^\circ\-\sin60^\circ&\cos60^\circ\end{pmatrix}$, a rotation matrix by $-60^\circ$ (i.e. $\alpha=60^\circ=\pi/3$), so $A^n$ is rotation by $-n	imes60^\circ$.

$A^{30}$: rotation by $-1800^\circ\equiv0^\circ \pmod{360^\circ}$, so $A^{30}=I$.

$A^{25}$: rotation by $-1500^\circ\equiv-60^\circ\pmod{360^\circ}$, so $A^{25}=A$.

$A^{30}+A^{25}-A=I+A-A=I$.

Solutions · 1