First, we'll learn how to make a coefficient matrix. We'll start with the following examples:
x + 4y = 6
2x – 3y = 9
This is all fine and dandy; we know all about coefficients. The first equation has a coefficient of 4, while the second equations has a 2 and a -3. But what's the coefficient in front of the first x? That's right; it's a 1. Therefore, our equations are equivalent to:
1x + 4y = 6
2x – 3y = 9
To create the coefficient matrix, we make a matrix like this:
With a coefficient matrix, the coefficients are the entries. And don't forget, the rows of a matrix are the horizontal number groups, and the columns are the vertical groups (like columns that hold up a ceiling).
Now we get to find us some determinants.
The determinant is a special value that we can pull out of a square matrix. Determinants only associate with square matrices (insecurity, no doubt), so you can't find the determinant of a matrix that's not a square.
To find the determinant from our coefficient matrix above, you go like this:
(1)(-3) – (2)(4) = -5
Why? Because you multiply down the first diagonal:
And up the other one:
And put a minus between them. That's it. Crisscross minus applesauce.
We also ditch the brackets and use vertical lines instead when we're talking about determinants. So the determinant of matrix X is written as |X|.
Looking for a formula, you math lovers? Okay.
= ps – rq
Important Shmoop Note: the whole crisscross-minus-applesauce thing only works with 2 × 2 matrices. Finding the determinant gets more complicated when we're dealing with 3 × 3 matrices and bigger, but we'll get into that a little later.
Craving more fun with determinants? Check out Cramer's Rule next.