Meaning of signed volume I want to understand the definition of the determinant of a $n\times n$ real matrix $A$ as the signed volume of the image of the unit cube $C'$ under the linear transformation given by $A$, i.e. $x\to Ax$. However I am failing to make sense of the words signed volume. What will be the precise definition of this? Will it be the $\int\int...\int dx_1dx_2\cdots dx_n$ over the image $C'$. I think this is an overcomplicated way to define the signed volume. Can someone suggest another way?
Thanks

In this context, signed volume is simply a term that carries slightly more information than volume alone. It's analogous to speed and velocity. 
The magnitude of the determinant of a linear transformation is the number that it scales volumes in the space by. We only need to consider the unit ball however, because if you know it scales the unit ball by that number, by linearity we can simply multiply arbitrary volumes by the same number and it will tell us the volume of the scaled volume for the arbitrary figure. 
However, we don't need to take the magnitude of the determinant, we can simply work with the determinant. In fact, taking the magnitude is losing some information; namely the sign. The sign actually tells us something interesting: it tells us whether the linear transformation inverts the space. To understand inversion, we can tell if a linear operator has inverted our space if we can't rotate and scale our way back. Think about the plane and the two operators: 
\begin{bmatrix}
2 & 0 \\ 
0 & 2 \end{bmatrix}
\begin{bmatrix}
-2 & 0 \\ 
0 & 2 \end{bmatrix}
The first one scales any volume by 4. The second one has determinant -4, but if we take magnitudes we see that it also scales volumes by 4. The difference is that volumes have been inverted under the second operator. We can tell it's inverted because even if we re-scale by $1/2$, we cannot rotate our way back to our original coordinate system (We cannot rotate the coordinate system $(-\hat{x},\hat{y})$ to $(\hat{x},\hat{y}))$.
To put it simply, we have the following complete notion:
For a linear operator 
$T : V \to V$, $A \subset V$, $B=T(A) $,
$Vol(B) = Vol(A)*|det(T)|$
If $det(T) < 0$, then $T$ inverts the space. If not, $T$ does not invert the space. 
The fact that the determinant gives us this extra data of invertedness is the reason why calling it just volume would be a little unfair, so we call it signed volume.
