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:
(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.
Create a coefficient matrix from these equations:
3x – 5y = 4
What are the rows and columns in the matrix:
Find D, the determinant of the coefficient matrix from Example One above:
Find D from this matrix: