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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:


(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.

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 = 0-x – 5y = 4

Exercise 2

Create a coefficient matrix:

3x – 4y = -7x + y = 9

Exercise 3

Create a coefficient matrix:

2x + 2y = 116xy = 3

Exercise 4

Find the determinant:

Exercise 5

Find the determinant:

Exercise 6

Find the determinant:

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