In ex 9, page 60, he writes down that in order to prove that each invertible matrix $A$ can be written as $A=e^X$, where $X\in M_{n\times n}$, one need to use the fact that if $A$ is unipotent then $\exp (\log A) = A$ and the fact that $A$ is similar to a block diagonal matrix where in the diagonal there are matrices of the form $\lambda I+ N_{\lambda}$, where $N_{\lambda}$ is a nilpotent matrix.
Obviously after I'll prove that $A$ is unipotent then I only need to take $X=\log A$.
I am not sure how to show this, I need to show that $A-I$ is nilpotent. I thought of taking the maximum integer, $\max\{k_1,...,k_n\}$, where $N_{\lambda_j}^{k_j} =0$, and $k_j$ is the minimal that satisfies this.
Don't know how to show this?
Any hints? Thanks.