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.

## Practice:

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

A matrix is just a grid of numbers inside brackets. When it's a coefficient matrix the coefficients are the entries. The coefficient matrix here is: | |

What are the rows and columns in the matrix:
| |

The rows of a matrix are the horizontal number groups and the columns of a matrix are the vertical number groups, so in this coefficient matrix the rows and columns are: Row one = 3*x* -5*y*, row two = -2*x* *y*: column one = 3*x* -2*x*, column two = -5*y* *y*. | |

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

To find D you multiply down the first diagonal And up the second And put a minus between them. Crisscross minus applesauce. For this example: D = (3*x*)(*y*) – (-2*x*)(-5*y*) | |

Find D from this matrix: | |

To find D, a determinant, we put the matrix in bars rather than brackets: We know thanks to the formula that to find D you multiply down the first diagonal and up the second, and then put a minus between them. That's our crisscross applesauce thing (no jump rope needed): D = (2)(1) – (3)(2) = 2 – 6 = -4 | |

Create a coefficient matrix:

3*x* + *y* = 0-*x* – 5*y* = 4

Hint

Coefficients are the numbers in front of the variables, and the answer will be 2 by 2 matrix.

Answer

Create a coefficient matrix:

3*x* – 4*y* = -7*x* + *y* = 9

Hint

Coefficients are the numbers in front of the variables, and the answer will be 2 by 2 matrix.

Answer

Create a coefficient matrix:

2*x* + 2*y* = 116*x* – *y* = 3

Hint

Coefficients are the numbers in front of the variables, and the answer will be 2 by 2 matrix.

Answer

Find the determinant:

Hint

Crisscross minus applesauce.

Answer

Find the determinant:

Find the determinant:

Hint

Crisscross minus applesauce.

Find the determinant:

Hint

Crisscross minus applesauce.