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

A matrix defines a linear transformation that changes not only the volume and the shape of objects in space, but also their orientation. The magnitude of the determinant tells you by how much the volume of the object is scaled up / down. The sign of the determinant tells you whether the orientation of the object is preserved (positive sign) or changed (negative sign). A graphical illustration of these concepts can be found at https://www.statlect.com/matrix-algebra/determinant-of-a-matrix. 
