1. ## need to proof

Hi, I need to prove this statement, but my teacher wants me to so without using $\displaystyle adj{A}$ and I kind of stuck because using Cramer's rule to prove that is almost like using the $\displaystyle adj{A}$.
"$\displaystyle A\in M_n(\mathbb{Q})$ therefore $\displaystyle A\in M_n(\mathbb{R})$
prove that: if $\displaystyle A$ is nonsingular in$\displaystyle M_n(\mathbb{R})$, $\displaystyle A$ is also nonsingular in $\displaystyle M_n(\mathbb{Q})$"

2. ## Re: need to proof

Can you show that $\displaystyle A^{-1}$ has only rational entries?

3. ## Re: need to proof

Originally Posted by xbox360
Hi, I need to prove this statement, but my teacher wants me to so without using $\displaystyle adj{A}$ and I kind of stuck because using Cramer's rule to pr ove that is almost like using the $\displaystyle adj{A}$.
"$\displaystyle A\in M_n(\mathbb{Q})$ therefore $\displaystyle A\in M_n(\mathbb{R})$
prove that: if $\displaystyle A$ is nonsingular in$\displaystyle M_n(\mathbb{R})$, $\displaystyle A$ is also nonsingular in $\displaystyle M_n(\mathbb{Q})$"
I assume that $M_n(\mathbb{R})$ is the set of all $n\times n$ matrices having real entries. If that is correct then $M_n(\mathbb{Q}) \subset M_n(\mathbb{R})$. Any nonsingular square matrix has non-zero determinate. Can you explain what you are and are not allowed to do for a proof.

4. ## Re: need to proof

One way to do this is to transform the problem to vector spaces. Here's the relevant theorem:

Let V be a vector space of finite dimension n over a field F and T a linear transformation from V to V. Then $T^{-1}$ exists if and only if Ker(T)=0. If $T^{-1}$ exists, $T^{-1}$ is a linear transformation.

You can use this theorem to prove your statement (without determinants at all). If you don't see how to do this or need help on the theorem, post again.

5. ## Re: need to proof

Originally Posted by xbox360
Hi, I need to prove this statement, but my teacher wants me to so without using $\displaystyle adj{A}$ and I kind of stuck because using Cramer's rule to prove that is almost like using the $\displaystyle adj{A}$.
"$\displaystyle A\in M_n(\mathbb{Q})$ therefore $\displaystyle A\in M_n(\mathbb{R})$
prove that: if $\displaystyle A$ is nonsingular in$\displaystyle M_n(\mathbb{R})$, $\displaystyle A$ is also nonsingular in $\displaystyle M_n(\mathbb{Q})$"
@xbox360, why do you not answer the replies to your post? Truth be told, what exactly is the question?

Is it $\displaystyle A$ is nonsingular in$\displaystyle M_n(\mathbb{R})$ then $\displaystyle A$ is also nonsingular in $\displaystyle M_n(\mathbb{Q})~?$

Or is it $\displaystyle A\in M_n(\mathbb{Q})$ and is nonsingular in$\displaystyle M_n(\mathbb{R})$, then $\displaystyle A$ is also nonsingular in $\displaystyle M_n(\mathbb{Q})~?$