Question

Question: Find the order of the given element.

${\left(c\right)}^{\left(\begin{array}{cc}-\frac{1}{2}& \frac{1}{2}\\ -\frac{3}{2}& -\frac{1}{2}\end{array}\right)}$in $\mathit{G}\mathit{L}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{ℝ}\mathbf{)}$

Short answer

The order of ${}^{\mathbf{\left(}\begin{array}{cc}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}& \frac{\mathbf{1}}{\mathbf{2}}\\ \mathbf{-}\frac{\mathbf{3}}{\mathbf{2}}& \mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\end{array}\mathbf{\right)}}$in $GL(2,\mathrm{ℝ})$ is$3$.

Step-by-step solution

Step 1: GL(2,ℝ)

The group$\mathit{G}\mathit{L}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{ℝ}\mathbf{)}$ is a group of all $\mathbf{2}\mathbf{\times }\mathbf{2}$invertible matrices over the field of real numbers $\mathbf{ℝ}$

Order of an element

Order of an element $\mathit{a}\mathbf{\in }\mathit{G}$is defined as a positive integer n such that ${\mathit{a}}^{\mathbf{n}}\mathbf{=}\mathit{e}$,where $\mathit{e}$is the identity element in $\mathit{G}$.

Conclusion

The given element in $GL(2,\mathrm{ℝ})$is ${}^{\left(\begin{array}{cc}-\frac{1}{2}& \frac{1}{2}\\ -\frac{3}{2}& -\frac{1}{2}\end{array}\right)}$. Now

$$\begin{gathered} {\left(\begin{array}{cc}-\frac{1}{2}& \frac{1}{2}\\ -\frac{3}{2}& -\frac{1}{2}\end{array}\right)}^2=\left(\begin{array}{cc}-\frac{1}{2}& \frac{1}{2}\\ -\frac{3}{2}& -\frac{1}{2}\end{array}\right)\left(\begin{array}{cc}-\frac{1}{2}& \frac{1}{2}\\ -\frac{3}{2}& -\frac{1}{2}\end{array}\right)=\left(\begin{array}{cc}-\frac{1}{2}& -\frac{1}{2}\\ \frac{3}{2}& -\frac{1}{2}\end{array}\right), \\ {\left(\begin{array}{cc}-\frac{1}{2}& \frac{1}{2}\\ -\frac{3}{2}& -\frac{1}{2}\end{array}\right)}^3=\left(\begin{array}{cc}-\frac{1}{2}& \frac{1}{2}\\ -\frac{3}{2}& -\frac{1}{2}\end{array}\right)\left(\begin{array}{cc}-\frac{1}{2}& -\frac{1}{2}\\ \frac{3}{2}& -\frac{1}{2}\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right) \end{gathered}$$

So, the order of${}^{\left(\begin{array}{cc}-\frac{1}{2}& \frac{1}{2}\\ -\frac{3}{2}& -\frac{1}{2}\end{array}\right)}$ in $GL(2,\mathrm{ℝ})$is$3$.