In two dimensional space there is a simple formula for the area of a parallelogram bounded by vectors v and w with v = (a, b) and w = (c, d): namely ad - bc
Why is this so?
1. The statement that the area of a parallelogram with sides given by the vectors (a, b) and (c, d) is |ad - bc| is obviously true if b and c are 0, since the parallelogram is then a rectangle with sides |a| and |d|, whose area is |ad|.
2. Neither the area nor ad - bc changes if we add a multiple of (a, b) to (c, d) or vice versa. The parallelogram merely tilts, and its base and altitude remain the same. If we for example, add a to c, we get a change of ad - bc to ad - b(c + a) so of -ba; but adding b to d produces a compensating change of ab to it; a(b + d) - b(c + a) = ad - bc and the net change is 0.
3. Starting with any a, b, c and d, by repeatedly adding multiples of one row to another we can force b and c to be 0, after which we are in the case considered in the first paragraph and we know that the area is |ad - bc|.
4. Since these addings didn't change the area and didn't change |ad - bc|, these must have been the same from the start.
The combination ad - bc is called the determinant of the matrix
whose rows are (a,b) and (c,d). It is often written as
Given three vectors in three dimensions we can form a 3 by 3 matrix of their components, and we will see that the absolute value of the determinant of that matrix is the volume of the parallelepiped whose edges are determined by the three vectors.
In fact an analogous results holds for k k-vectors: the absolute value of the determinant of the matrix of their components is the k-volume of the figure they bound.
What then is the determinant of a matrix?
In any dimension, it is defined as follows:
1. A determinant is linear in the elements of any row (or column) so that multiplying everything in that row by z multiplies the determinant by z, and the determinant with row v + w is the sum of the determinants otherwise identical with that row being v and that row being w.
2. It changes sign if two of its rows are interchanged (an equivalent condition is that it is 0 if two rows are identical).
3. The matrix with 1's on the diagonal and 0's elsewhere has determinant 1.
An alternate definition of the determinant of a k by k matrix is that it is the "signed" and therefore linear version of volume in k dimensions.
The determinant is generally written as det M or as | M | or sometimes || M ||.
If v and w are two rows of the matrix M we can deduce from the first two conditions that adding a multiple of v to w does not change the determinant of M.
The volume of a parallelepiped bounded by edges whose directions and lengths are that of u, v and w is almost linear in u, v and w; it differs from linearity only in that it is always positive, like length in one dimension is.
This volume is 1 if the vectors have unit length and are mutually perpendicular, and does not change if one side is added to another; (that just tilts the parallelepiped without changing its volume.) The absolute value of the determinant of the matrix formed by the components of the three vectors obeys exactly the same conditions and is therefore the same thing.
In higher dimensions the analog of volume is called hyper-volume and the same conclusion can be drawn by the same argument: the hyper-volume of the parallel sided region determined by k vectors in k dimensions is the absolute value of the determinant whose entries are their components in the directions of the (orthonormal) basis vectors.
In fact, the determinant can be considered a linear and signed version of hyper-volume.
Consider the hyper-volume of a parallel-sided region with sides xA, B, C, ..., as a function of the variable x. It is linear in x for positive or negative x, but it is always positive, and its graph looks like a V, taking the value 0 for x = 0.
The determinant is the same as the hyper-volume for positive or negative x and minus the hyper-volume for the other, and is linear as a function of x. Its sign is determined by the convention that it is positive for the "identity matrix" which has 1's on the main diagonal and 0's elsewhere. This identity matrix is usually written as I.
4.0 Prove the statement above: the condition on the determinant that it change sign if two rows are interchanged is equivalent to the alternative condition that the determinant is 0 if two rows are identical (given its linearity in rows).
4.1 Suppose A, B and C are three vectors in the plane. Consider three triangles with sides A and B, A and C, and A and B + C, respectively. What relation holds between their areas? (If you don't see it, try some simple examples and generalize. The relation sought here is an either or statement.) What is the analogous statement about the determinants whose rows are the components of the given vectors (A and B), (A and C) and (A and B + C)?
In the following applet you can enter three 3-vectors, see them and the parallelepiped they define, and the value of the determinant whose absolute value is its volume. We will soon see how to compute the determinant. Also shown are the vector (or cross) products of pairs of these vectors which will be defined in section 4.5.