Consider a matrix A where det(A) = 18. What does this value 18 actually represent?

Answer 1

The determinant of an NxN matrix is the "size" of the shape the matrix describes.

For example, the determinant of the 1x1 matrix
[k]
is simply k. Imagine the first (and only row) of the matrix as a vector with one dimension. What is the size of the matrix? Well, it's the length of that one-dimensional vector.

The determinant of a 2x2 matrix
[a,b]
[c,d]
is (a*d - b*c). Imagine that each row of the matrix represents a vector with two dimensions. If you construct a parallelogram using the two vectors as the sides, the determinant is equal to the area of that parallelogram.

The determinant of a 3x3 matrix is simply a further extension of this idea. If you take three 3D vectors and construct a parallelpiped
http://en.wikipedia.org/wiki/Parallelepiped
the determinant of the matrix will be the volume of the parallelpiped.

It goes into progressively higher dimensions with larger matrices.

If the determinant is negative, it simply means the parallel object is in a particular quadrant (or octant or whatever), as opposed to another.

Answer 2

There are two very important points:
First, note that the determinant function is only defined when A is a square matrix. A square matrix maps vectors in R^n to other vectors in R^n. That means that determinants are applicable specifically to the case in which a linear transformation A is mapping from one set of basis vectors to another set of basis vectors: i.e., a change of coordinates.

1) If det(A) = 0, that means that at least one dimension of the original basis vectors has been lost. This happens if you're mapping a 3-dimensional space into a plane or a line: you lose a dimension or two, and it shows up by zeroing out the determinant. (It also applies for higher dimensions.)

2) If det(A) not = 0, the absolute magnitude of det(A) tells what happens to an n-dimensional volume that is transformed by A. In other words, take n vectors X1, X2, .. Xn, in the original space, and calculate the n-dimensional volume they define. Now, map each of those n vectors onto its image, A*X1, A*X2, .. A*Xn. These new n vectors also define an n-dimensional volume. The ratio between these volumes is det(A).

So in your specific case, the transformation A will map an n-dimensional volume into another n-dimensional volume - but it will be 18 times as great.

(If det(A) is negative, it means there is an inversion. For most purposes, this is not too important.)

This turns out to be useful when you are converting from cartesian to curvilinear coordinates. In that case, the amount of volume expansion/compression varies from point to point, so if you look at the mappings between dx, dy, dz and dr, d(theta), d(phi), the coefficients vary, and the local determinant varies. The local determinant is called the "Jacobian". The fact that it varies from point to point, and the fact that this relates to the expansion/compression factor, explains why when you go from cartesian to spherical coordinates, the integration changes from:
dx dy dz
to
[r^2 sin(theta)] dr d(theta) d(phi)

and when you go from cartesian to cylindrical coordinates, it goes from:
dx dy dz
to
[r] dr d(theta) dz

The stuff in the [] comes from the Jacobian.

Answer 3

If the matrix is a 2x2 then it is the area scale factor when the matrtix is applied to a shape.

In a 3x3 matrix it will give you the volume scale factor.

After that who knows what it gives you!!!!!

Answer 4

The determinant represents the "magnitude" of the matrix, since the inverse matrix has the reciprocal of the determinant.

Answer 5

In and of itself, the determinant value of 18 tells you nothing. It is what you do with that value afterwards that determines it's worth.

Essentially, a determinant simply helps you to go onto the next step.

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Edit:

It is important to know who your audience is. A first time poster making a simple request regarding the determinant value almost assuredly doesn't want a thesis explanation. Sometimes being terse and concise is more effective in what they want to know.

Answer 6

it's not clear what you ask
if you want det is a number that "measures" the relation between the columns or lines of the matrix