We have changed our privacy policy. In addition, we use cookies on our website for various purposes. By continuing on our website, you consent to our use of cookies. You can learn about our practices by reading our privacy policy.

# Matrices

Beginning Operations

# 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 -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:

That's all a matrix really is: a grid of numbers inside brackets. When it's a coefficient matrix, the coefficients are the entries. The rows of a matrix are the horizontal number groups, so in this coefficient matrix the rows are

1 4 and 2 -3; 1 4 is row one, and 2 -3 is row two.

The columns of a matrix are the vertical number groups, so in this coefficient matrix the columns are

1 2 and 4 -3; 1 2 is column one and 4 -3 is column two.

Now we get to find us some determinants.

Determinants ditch the brackets and use vertical lines instead. Determinants are NOT open-minded. They only associate with square matrices. (Insecurity, no doubt.)

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.

Looking for a formula, you math lovers? Okay.

The value of the determinant for the above matrix is

ps – rq

Fun with determinants doesn't end there. Check out Cramer's Rule next.

#### Example 1

 Create a coefficient matrix from these equations:3x – 5y = 4-2x + y = 5

#### Example 2

 What are the rows and columns in the matrix:

#### Example 3

 Find D, the determinant of the coefficient matrix from Example One above:

#### Example 4

 Find D from 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: