# Given characteristic polynomial of $T$, need find characteristic polynomial of $T^3$

Let $T:\mathbb{R}^2\to \mathbb{R}^2$ be a linear transformation with characteristic polynomial $x^2+2x-3$. Find the characteristic polynomial of $T^3$.

How to do this?

Thanks!

• I'm guessing you mean $T: \mathbb{R}^2 \to \mathbb{R}^2$. Do you know the relation between the eigenvalues of $T$ and $T^3$? – Pedro M. Jul 26 '13 at 13:20
• Yes, You are correct! – 17SI.34SA Jul 26 '13 at 13:23

HINT: If $\lambda$ is an eigenvalue of $\operatorname{T}$ then $\lambda^n$ is an eigenvalue of $\operatorname{T}^n$.
Note that the transformation matrix of $T$ is symmetric with$\lambda=1,-3$. Hence the eigenvalues of the transformation matrix of $T^3$ are $1,-27$. So the characteristic equation is $$(x-1)(x+27)=0$$