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True or false?

For any $n ≥ 2$$n ≥ 2$, there exists a $latex n \times n$ real matrix $latex A$ such that the set $latex \{A^p \mid p ≥ 1\}$ spans the $latex \Bbb R$-vector space $latex M_n(\Bbb R)$.

Source of the problem
TIFR GS 2019, Part B Problem 6
Easy
Suggested Book
Linear algebra, Friedberg, Insel

Do you really need a hint? Try it first!

Can you use Cayley Hamilton theorem which states that a $$n \times n$$ matrix $$A$$ will satisfy it’s characteristic polynomial of degree $$n$$.

This is false since Cayley Hamilton implies that $dim(V=Sp\{A^p, p\in\mathbb{N}\})\leq n+1$ and $dim(M_n(\mathbb{R})=n^2$.

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