# Find the units of a ring!

I am given the following exercise: Find the units of the ring $R=M_n(\mathbb{Q})$.

That's what I have thought:  $E \in M_n(\mathbb{Q})$ is an unit of the ring $R=M_n(\mathbb{Q})$,if $\exists$ $E' \in M_n(\mathbb{Q})$ such that $EE'=E'E=I_n$,that means that $E$ must be inversible.So,$det(E) \neq 0$.  How can I continue or is that the answer?

• That's the answer; because of the explicit expression of the inverse by means of minors of $E$, the inverse of a matrix with rational coefficients has rational coefficients. Mar 30 '14 at 13:29

That is to say, it's certainly true that every unit in $M_n(\mathbb Q)$ has nonzero determinant. But is every element with nonzero determinant a unit?