## Some properties of a QQ-vector space given the existence of a linear transformation with given properties

Let $V$ be a finite dimensional vector space over $\mathbb{Q}$ and suppose $T$ is a nonsingular linear transformation on $V$ such that $T^{-1} = T^2 + T$. Prove that the dimension of $V$ is divisible by 3. If the dimension of $V$ is precisely 3, prove that all such transformations are similar.

If $T$ is such a transformation, then we have $1 = T^3 + T^2$, and so $T^3 + T^2 - 1 = 0$. So the minimal polynomial of $T$ divides $p(x) = x^3 + x^2 - 1$. Now $p(x)$ is irreducible over $\mathbb{Q}$ by the Rational Root Test (Prop. 11 on page 308 in D&F). So the minimal polynomial of $T$ is precisely $p(x)$. Now the characteristic polynomial of $T$ divides some power of $p(x)$ (Prop. 20 in D&F) and so has degree $3k$ for some $k$. But of course the degree of the characteristic polynomial is precisely the dimension of $V$, as desired.

Now if $V$ has dimension 3, then the minimal polynomial of $V$ is the characteristic polynomial, so that the invariant factors of $T$ are simply $p(x)$. In particular, all such $T$ are similar.