# Beginning Operations

First, we'll learn how to make a **coefficient matrix**. We'll start with the following examples:

*x* + 4*y* = 6

2*x* – 3*y* = 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:

1*x* + 4*y* = 6

2*x* – 3*y* = 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:

becomes

(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.