At a Glance  Beginning Operations
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:
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 crisscrossminusapplesauce 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.
Example 1
What are the rows and columns in the following matrix?

Example 2
Find D, the determinant of the following matrix: 
Example 3
What's the determinant of this matrix? 
Exercise 1
Create a coefficient matrix:
3x + y = 0x – 5y = 4
Exercise 2
Create a coefficient matrix:
3x – 4y = 7x + y = 9
Exercise 3
Create a coefficient matrix:
2x + 2y = 116x – y = 3
Exercise 4
Find the determinant:
Exercise 5
Find the determinant:
Exercise 6
Find the determinant: