NO LINKS OR ANSWERING QUESTIONS YOU DON'T KNOW!! Part 2. 2. What is a determinant and what role does it play with matrices (Hint: What does a determinant of 0 mean)? How can this be used when solving systems of equations?

Let's look at an example.

If we wanted to solve this system of equations

Then it's equivalent to solving this matrix equation

We can then further condense that into the form

Where,

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To solve the matrix equation Aw = B, we could compute the inverse matrix and left-multiply both sides by this to isolate w.

So we'd go from to . The order of multiplication is important.

For any 2x2 matrix of the form

its inverse is

Notice the expression ad-bc in the denominator of that fractional term outside. This expression represents the determinant of matrix P. Some books may use the notation "det" to mean "determinant"

or you may see it written as

Those aren't absolute value bars, even if they may look like it.

Based on that, we can see that the determinant must be nonzero in order to compute the inverse of the matrix. Consequently, the determinant must be nonzero in order for Aw = B to have one solution.

If the determinant is 0, then we have two possibilities:

So a zero determinant would have to be investigated further as to which outcome would occur.

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Let's return to the example and compute the inverse (if possible).

In this case, the inverse does exist.

This further leads to

This shows that the solution is (x,y) = (3,4).

As the other person pointed out, you could use Cramer's Rule to solve this system. Cramer's Rule will involve using determinants and you'll be dividing over determinants. So this is another reason why we cannot have a zero determinant.