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|

translates into:

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

which implies:

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.