Question

For the matrix A in exercise 33 through 42 , Compute $A^2=AA,A^3=AAA$, and $A^4$. Describe the pattern that emerges, and use this pattern to find $A^{1001}$. Interpret your answers geometrically, in terms of rotations, reflections, shears, and orthogonal projections.

$A=\frac{1}{2}\left[\begin{array}{cc}-1& -\sqrt{3}\\ \sqrt{3}& -1\end{array}\right]$

Short answer

$A^{1001}=\frac{1}{2}\left[\begin{array}{cc}-1& \sqrt{3}\\ -\sqrt{3}& -1\end{array}\right]$

And the matrix describes a rotation through angle $120°$in the counter clockwise direction.

Step-by-step solution

Calculate

Let

$A=\frac{1}{2}\left[\begin{array}{cc}-1& \sqrt{3}\\ -\sqrt{3}& -1\end{array}\right]$

Then we have

$$\begin{gathered} A^2=AA \\ =\frac{1}{2}\left[\begin{array}{cc}-1& -\sqrt{3}\\ \sqrt{3}& -1\end{array}\right]\left[\begin{array}{cc}-1& -\sqrt{3}\\ \sqrt{3}& -1\end{array}\right] \\ =\frac{1}{2}\left[\begin{array}{cc}-1& \sqrt{3}\\ -\sqrt{3}& -1\end{array}\right] \end{gathered}$$

$$\begin{gathered} A^3=A^{2}A \\ =\frac{1}{2}\left[\begin{array}{cc}-1& -\sqrt{3}\\ -\sqrt{3}& 1\end{array}\right]\frac{1}{2}\left[\begin{array}{cc}-1& -\sqrt{3}\\ -\sqrt{3}& 1\end{array}\right] \\ =\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right] \end{gathered}$$

$$\begin{gathered} A^4=A^{3}A \\ =I_{2}A \\ =A \end{gathered}$$

Clearly,, $A^{999}=I_2$as $999=333\times 3$and $A^3=I_2$.

Hence we get

$$\begin{gathered} A^{1001}=A^{999}{A}^2 \\ =A^2 \\ =\frac{1}{2}\left[\begin{array}{cc}-1& \sqrt{3}\\ -\sqrt{3}& -1\end{array}\right] \end{gathered}$$

Representation of matrix A in terms of rotations, reflections, and projections

Clearly,

$A=\left[\begin{array}{cc}cos\left(2\pi /3\right)& -\sin \left(2\pi /3\right)\\ \sin \left(2\pi /3\right)& cos\left(2\pi /3\right)\end{array}\right]$

So the matrix A describes a rotation by $120°=\raisebox{1ex}{$2\pi $}\!\left/ \!\raisebox{-1ex}{$3$}\right.$in the counter clockwise direction.

The Final Answer

Therefore,

$A^{1001}=\frac{1}{2}\left[\begin{array}{cc}-1& \sqrt{3}\\ -\sqrt{3}& -1\end{array}\right]$

And the matrix describes a rotation through angle$120°$ in the counter clockwise direction.