# At a Glance - 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 -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:

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: 3 |

#### 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:

3*x* + *y* = 0-*x* – 5*y* = 4

#### Exercise 2

Create a coefficient matrix:

3*x* – 4*y* = -7*x* + *y* = 9

#### Exercise 3

Create a coefficient matrix:

2*x* + 2*y* = 116*x* – *y* = 3

#### Exercise 4

Find the determinant:

#### Exercise 5

Find the determinant:

#### Exercise 6

Find the determinant: