# Understand the problem

True or false?

### For any , 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

##### Topic

##### Difficulty Level

Easy

##### Suggested Book

Linear algebra, Friedberg, Insel

# Start with hints

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 and .