linear algebra – How to prove iff about diagonalizability and determinants?

I have the following linear algebra proof:

Question: Let $V$ be an n-dimensional vector space, $lambda in mathbb{R}$, and $T : V rightarrow V$ a linear map with det($T − tI$) = ($lambda − t)^n$. Show that $T$ is diagonalizable if and only if $T = λI$.

I’m not entirely sure where, to begin with, this, in particular how to use the fact that det($T − tI$) = ($lambda − t)^n$. How does that relate to diagonalizability?