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**deniselim17** Let $\displaystyle A \in M_{n}(mathbb{R})$.

If $\displaystyle A$ is invertible, show that $\displaystyle A^{n} \neq 0$ for all positive integer $\displaystyle n$.

I started with $\displaystyle n=1$, then $\displaystyle A^{n}=A^{1} \neq 0$ since $\displaystyle A$ is invertible.

Assume that $\displaystyle A^{n} \neq 0$ for some positive integer $\displaystyle n$.

Now, how to show that $\displaystyle A^{n+1} \neq 0$???

$\displaystyle A^{n+1}=A^{n}A^{1}$ but, even though $\displaystyle A^{n}, A^{1}$ nonzero matrix, $\displaystyle A^{n+1}$ can be zero matrix also.

I'm stuck here.