Matrix algebra can seem very abstract at times and what you are doing is often not very clear. Determinants, eigenvalues, eigenvectors and matrix multiplication are all very confusing.
Today I am going to try and give an intuitive example for a reason for using the determinant.
First we need to understand one of the purposes of matrix algebra. It is used to solve systems of linear equations. I will look at an example with a 2 x 2 linear system.
We could write it as Ax = M
which translates to:
|a b| |x| = |m|
|c d| |y| = |n|
ax + by = m
cx + dy = n
Now you should know that the formula for the determinant of A is:
DET(A) = a*d - cb
This is the most important thing to know. The value of the determinant does not mean anything unless it is 0. When it is 0 we get an interesting result.
If DET(A) = 0 we get:
a*d - cb = 0
which implies that:
a*d = c*b
a/b = c/d
which says that the slopes of the two equations are equal. This means that we cannot find a value for (x,y) where the two curves intersect because the never do!
If you cannot see why the slopes are equal, re-arrange the two systems as follows:
y = (m/b) - (a/b)*x
y = (m/d) - (c/d)*x
(which both look like our y = mx + c)
Thus the determinant is telling us that we can find a solution (or an intersection of two curves) for our system (which just means all our equations) when the value of the determinant does not equal 0.
There are other uses for the determinant, but this should hopefully provide some insight into the usefulness of the determinant.