Helpful Tools for Calculus, Chapter 2: Helpful Tools for the Mathematics of Calculus
Previous Section: 2.2 Proof
Systems of Equations
Once again, this topic is one that is deserving of its own text. We will focus on the use of matrices to solve a system of equations. We should start by introducing the context in which this is useful.
When we have a system of two or more equations there are multiple ways we can solve them to find the solution(s) that satisfy all of the equations at the same time. In middle school Algebra, the methods of substitution and graphing are often emphasized.
When solving a system with many equations it gets much more complicated and it is easy to organize with a matrix. A matrix is a structure that organizes the equations into rows and columns where each slot represents the coefficients of certain parts of each equation. Rows represent the equations while columns represent the terms:
x+2y+z=3
3x-y-3z=-1
2x+3y-z=4
This is a system of equations with three variables. That can be represented as:
The following notation is not often used, but is helpful when learning to understand what each part is:
The f’s are the functions in order and the columns represent the x, y, and z values as well as the constant. The vertical bar is most similar to the equal sign in the equations.
Example 1: Convert the following system into a matrix:
2x+3y+z=4
-x+5y+2z=-3
x+z=2
Solution:
Example 2: Convert the following system into a matrix:
x1-x2+x3=5
x1-x3=0
3x1-2x2=9
Solution:
Solving a Matrix
In order to solve a system of equations represented by a matrix, there are three main operations we can follow. These are called Elementary Row Operations and the process of using them is called Gaussian Elimination, named after the mathematician Carl Friedrich Gauss.
The Elementary Row Operations are as follows:
I. Any two rows can be interchanged
II. Any row can be multiplied by a nonzero real number
III. Any row can be added to another row
The goal is to make the matrix look like the following:
One of the better methods involves using the top row to cancel out the numbers in the first column. Using the row operations we simply combine the rows in a way that the result is a 0. We then use the same method to cancel out things in the second column, using the second row. This continues until we have a triangular matrix:
Then we should repeat the strategy, but in the opposite direction, using the bottom row to cancel out the right column, and so on.
Example 3: Solve the following matrix:
Solution:
The final form of this matrix, in what is called Reduced Row Echelon Form, gives us:
Thus, the solution to this matrix is x1=1, x2=1, x3=2. This can also be phrased as {1, 1, 2}.
Example 4: Solve the following matrix:
Solution:
The final form of this matrix, in what is called Reduced Row Echelon Form, gives us:
Thus, the solution to this matrix is x1=-3, x2=1, x3=2. This can also be phrased as {-3, 1, 2}.
Review
A matrix is used to represent a system of equations
Elementary Row Operations are used to solve a matrix:
Any two rows can be interchanged
Any row can be multiplied by a nonzero real number
Any row can be added to another row