Question: We know that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as additive groups(as they are isomorphic as $\mathbb{Q}$ vector spaces) under the axiom of choice. But why not $(\mathbb{Q}, +)$ and $(\mathbb{Q}^2,+$) isomorphic as additive groups?

Specifically, I think for vector space isomorphism, I think since they are the smallest field to begin with(they are field of fractions of the integers), so they do not have a common basis, hence may be they are not isomorphic as vector spaces. But, what about group isomorphism? I think there is a simple counterexample. Any hints? Thanks beforehand.

Answer: 
> 
> I can't figure out why the determinant changes, but there must be some relation since they're multiples of 370?
> 
> 
> 

Adding a scalar multiple of one row (or column) to *another* row (or column) does not change the determinant so in your third attempt, you can indeed add twice the second row to the first row and you obtain the correct value of $370$.

However, in your first and second attempt, you change the second row not only by adding another row (the first), but also the second row itself. This changes the determinant: multiplying a row (or column) with a factor $k$, results in the determinant being multiplied by that same factor $k$ as well, hence the $740$ (2 times $R\_2$ ) and $1110$ (3 times $R\_2$).

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As a side note, I would focus on the last column:
$$\begin{vmatrix}
4 & 5 & \color{red}{-3} \\ -2 & 21 & \color{blue}{1} \\ 8 & 7 & \color{red}{-2}
\end{vmatrix}
\underset{R\_3 \to R\_1+2R\_2}{\overset{R\_1 \to R\_1+3R\_2}{\longrightarrow}}
\begin{vmatrix}
-2 & 68 & \color{red}{0} \\ -2 & 21 & \color{blue}{1} \\ 4 & 49 & \color{red}{0}
\end{vmatrix}= - \left( -2 \cdot 49 - 4 \cdot 68 \right) = 370
$$
The determinant should be -74. 

$4\*[(-4)(-2) - (-1)(1)]$ $+$ $(-2)\*[(-2)(-2) - (8) (1)]$ $+$ $(-3)\*[(-2)(-1) - (8)(-4)]$ = $36 - 8 - 102 $ $=$ $-74$

Then the matrix modifications at each stage will scale the determinant by the factor applied to any row/column (Since $Det(c\*A) = c^3 A$ ; assuming A is a 3x3 matrix)