I am stuck for weeks with the following problem:
Let $A$ and $x$ be $n \times n$ and $n \times 1$ matrices, respectively, with all entries real and strictly positive. Assume that $A^2 x = x$. Show that $A x = x$.
This was on the first problem set on a course of linear algebra based on the book written by Hoffman & Kunze. We haven't seen eigenvalues and eigenvectors yet. So, while any solution that uses anything more advanced than the first 3 chapters of that book is welcome (it may incentivize me to study something!), it does not solve the problem.
Can anyone help? Thanks!
EDIT: My question was marked as a exact duplicate of a question by Igor Caetano Diniz. While that is the exact same question, that post has one wrong answer and one answer that has a theorem that I haven't studied yet. So it doesn't solve my problem.