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Motivation (geometric interpretation - but it seems that this text should have some images)
Determinants are a basic building block of [[linear algebra]], and are useful for finding areas and volumes of geometric figures, in [[Cramer's rule]], and in many other areas ways.
Determinants are a basic building block of [[linear algebra]], and are useful for finding areas and volumes of geometric figures, in [[Cramer's rule]], and in many other areas ways.
== Motivation ==
A matrix can be used to ''transform'' a geometric figure. For example, in the plane, if we have a triangle defined by its vertices ''(3,3), (5,1), and (1,4)'', and we wish to transform this triangle into the triangle of vertices ''(3,-3), (5,-9), and (1,2)'', we can simply do a matrix multiplication of each vertex by the matrix
<math>\begin{pmatrix}
1 & 0 \\
-2 & 1 \\
\end{pmatrix}</math>.
In this transformation, no matter what is the shape of the initial geometric figure, its position, or its area, the final geometric figure will have the same area and orientation.
It can be seen that matrix transformations of geometric figures ''always'' give resulting figures whose area is proportional to the initial figure, and whose orientation is either always the same, or always the reverse.
This ratio is called the '''determinant''' of the matrix, and it's positive when the orienation is kept, negative when the orientation is reversed, and zero when the final figure always has zero area.
This two-dimensional concept is easily generalized for any dimensions. In 3D, replace ''area'' for ''volume'', and in higher dimensions the analogue concept is called ''hypervolume''.
The '''determinant''' of a matrix is the oriented ratio of the hypervolumes of the transformed figure to the source figure.
==How to calculate==
==How to calculate==