What do the column of 1's signify when calculating the area of an image using the determinant? When calculating the area of an image such as that given by the three coordinates (4,3), (4,10), and (-4,-3), why when writing it out as a matrix is there an extra column of 1's?
$$
        \begin{matrix}
        4 & 3 & 1 \\
        4 & 10 & 1 \\
        -4 & -3 & 1 \\
        \end{matrix}
$$
I understand that it is not possible to find the determinant of a non-square matrix, so my questions are:
1) Are 1's just used to make the matrix a square?
2) If so, is there any reason why 1's and not 0's are used?
 A: There are two ways to look at the ones. 
Algebraically, we are using the properties of determinant to express the area of a triangle $T$ with vertices at $A : (x_1,y_1)$, $B : (x_2,y_2)$, $C : (x_3,y_3)$ in a simpler form:
$$\verb/Area/(T) = \frac12 
\left|
\begin{matrix}
x_2 - x_1 & y_2 - y_1\\
x_3 - x_1 & y_3 - y_1\\
\end{matrix}\right|
=
\frac12 
\left|
\begin{matrix}
0 & 0 & 1\\
x_2 - x_1 & y_2 - y_1 & 1\\
x_3 - x_1 & y_3 - y_1 & 1\\
\end{matrix}\right|
= \frac12
\left|
\begin{matrix}
x_1 & y_1 & 1\\
x_2 & y_2 & 1\\
x_3 & y_3 & 1\\
\end{matrix}\right|
$$
Geometrically, we can embed $\mathbb{R}^2$ as the plane $z = 1$ in $\mathbb{R}^3$. The vertices of $T$ becomes the points $A' : (x_1,y_1,1)$, $B' : (x_1,y_2,1)$, $C' : (x_3,y_3,1)$ on $\mathbb{R}^3$. 
Let $T'$ be the tetrahedron spanned by $A', B', C'$ and origin $O : (0,0,0)$.
The area of $T$ is 3 times the volume of tetrahedron $T'$.
The volume of  $T'$ is $\frac16$ of the volume of the parallelepiped $P$ with one vertex at $O$ and spanned by the 3 vectors $A', B', C'$. Since the volume of a parallelepiped can be expressed as a cross product which equals to corresponding determinant, we have:
$$\verb/Area/(T) = 3\verb/Volume/(T') = \frac{3}{6}\verb/Volume/(P)
= \frac12
\left|
\begin{matrix}
x_1 & y_1 & 1\\
x_2 & y_2 & 1\\
x_3 & y_3 & 1\\
\end{matrix}\right|
$$
In certain sense, the introduction of ones here reflect the possibility
to express geometric relations for objects living on the plane as geometric relations on $\mathbb{R}^3$. It allows one to look at plane geometry problem from a completely different angle and offer us new insight how to solve a problem.
