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.

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:

Example 2

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 = 3x -5y, row two = -2xy: column one = 3x -2x, column two = -5yy.

Example 3

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 = (3x)(y) – (-2x)(-5y)

Example 4

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):