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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 -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:


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