Let p be an odd prime number.

Let $A\in {M}_{n\times n}(\mathbb{Z})$ be a matrix satisfiying ${a}_{ij}\equiv {\delta}_{ij}\phantom{\rule{0.444em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}p)$

Prove that: if $|det(A)|=1$ and ${A}^{m}=I$ for some $m\in {\mathbb{N}}^{+}$, then we have $A=I$

Let $A\in {M}_{n\times n}(\mathbb{Z})$ be a matrix satisfiying ${a}_{ij}\equiv {\delta}_{ij}\phantom{\rule{0.444em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}p)$

Prove that: if $|det(A)|=1$ and ${A}^{m}=I$ for some $m\in {\mathbb{N}}^{+}$, then we have $A=I$