## 2.2 Using Matrices to Solve Systems of Equations

### Part A: Setting Up a System & Doing Row Operations

(Based on Section 2.2 in Finite Mathematics and Finite Mathematics and Applied Calculus)

Let us start by quickly reviewing some basic terms from the tutorial for Section 2.1.

Q Just what is a "system of linear equations in two unknowns?"

A First, a linear equation in two unknowns x and y is an equation of the form

ax + by = c

where a, b, and c are numbers, and where a and b are not both zero.

Examples: Linear Equations:

 4x + 5y = 0 This has a = 4, b = 5, c = 0 x - y = 11 This has a = 1, b = -1, c = 11 4x = 3 This has a = 4, b = 0, c = 11

Second, a system of linear equations is just a collection of these beasts. To solve a system of linear equations means to find a solution (or solutions) (x, y) that simultaneously satisfies all of the equations in the system.

Example: System of Linear Equations:

 4x + 5y = 40 x-y = 1 This is a system of two linear equations with solution x = 5, y = 4.

Setting Up a System of Linear Equations in Matrix Form

Simply put, the augmented matrix form of a single linear equation ax + by = c is just the single row matrix [a  b  c]. The augmented matrix of a whole system is then a matrix with one row for each equation in the system.

Examples: Matrix Form of a System:

System of Equations
Matrix Form
x - 2y = 5
3x         = 9
 1 -2 5 3 0 9
-3x + 2y = 10
2x - y = 0

-x - y + z = 1
2x - y       = 5

 2 - 1 0 4 -2 0 1 5

Doing Row Operations

Here are three things you can do to a system of equations without effecting the solution:

1. Switch any two equations
2. Multiply both sides of any equation by a non-zero number
3. Replace any equation by its sum with another equation. More generally, you can replace an equation by, say, 4 times itself plus 5 times another equation.

Corresponding to these changes are the following row operations on an augemented matrix.

Row Operation
Example
1. Switch two rows
We write R1R2 to indicate switching Row 1 and Row 2
 1 -2 5 3 0 9

R1R2
 3 0 9 1 -2 5
2. Multiply a row by a non-zero number

For instance, write the instruction
3 R2
next to Row 2 to mean "Multiply row 2 by 3."
 1 -2 5 3 0 9 3 R2
 1 -2 5 9 0 27
3. Replace a row by a combination with another row
For instance, write the instruction
3 R1-2 R2
next to Row 1 to mean:
"Replace Row 1 by three times Row 1 minus twice Row 2.
In words: "Three times the top minus twice the bottom."
 1 -2 5 3R1-2R2 3 0 9
 -3 -6 -3 3 0 9

Press here to see how we got that.

Q What is the effect of performing the following operation?
 -2 -2 0 -1 1 3 R2 + 2 R1
 -5 -3 3 -1 1 3
 -2 -2 0 3 5 3
 -2 -2 0 -5 -3 3

Now perform the indiated row operations and press "Check."

 -1 0 1 1 2R1 + R2 2 3 0 -1 4 -1 1 1 R3 - 2R2

You can now go on to the next part of the tutorial for Section 2.2 by pressing "Next Tutorial" on the sidebar.

Last Updated: March, 2006
Copyright © 1999, 2006 Stefan Waner and Steven R. Costenoble