Prove that $T$ is invertible if $(T\circ T)+T −2I_V = 0$

Question

I'm having difficulty proving the following:

Let $T:V \to V$ be a linear transformation such that $(T\circ T)+T −2I_V = 0$. Prove that $T$ is invertible.

I think that T might be equal to the identity transformation itself ($I$) but I'm not sure how to get to it from the datum.

Answer 1

$(T\circ T)+T −2I_V = 0$

$(T\circ T)+T =2I_V$

$\frac{1}{2} ((T\circ T)+T )=I_V$

$\frac{1}{2} T \circ(T+I_V) =I_V$

$T \circ\frac{1}{2}(T+I_V) =I_V$

so $\frac{1}{2}(T+I_V) =T^{-1}$