inverse of a matrix belonging to the same set

Question

Is it true that if $A$$\in M_{n\times n}$ then $A^{-1}\in M_{n\times n}$? $A$ is a matrix, I considered it as true but I cannot manage to prove it.

Answer 1

Suppose that we are working over a field $F$ and that $F'$ is an extension of $F$. Suppose furthermore that $A$ has an inverse in $M_{n\times n}(F')$. Does it follow that $A^{-1}\in M_{n\times n}(F)$? Yes, it does. For instance, you can use the fact that$$A^{-1}=\frac1{\det A}\operatorname{adj}(A),$$where $\operatorname{adj}(A)$ is the adjugate matrix of $A$. Since $\det A$ and each entry of $\operatorname{adj}(A)$ can be obtained from the entries of $A$ using only sums, subtractions, and products and since $F$ is a field, $A^{-1}\in M_{n\times n}(F)$.