Gaussian Elimination

Carl Friedrich Gauss lived during the late 18th century and early 19th century, but he is still considered one of the most prolific mathematicians in history. His contributions to the science of mathematics and physics span fields such as algebra, number theory, analysis, differential geometry, astronomy, and optics, among others. His discoveries regarding matrix theory changed the way mathematicians have worked for the last two centuries.

painting of Gauss German mathematician Carl Friedrich Gauss (1777–1855).


We first encountered Gaussian elimination in Systems of Linear Equations: Two Variables. In this section, we will revisit this technique for solving systems, this time using matrices.

Row Operations and the Augmented Matrix

A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix.

For example, consider the following
2×22\times 2
system of equations.

3x+4y=74x2y=5\begin{array}{l}3x+4y=7\\ 4x - 2y=5\end{array}
We can write this system as an augmented matrix:

[3442  75]\left[\begin{array}{rr}\qquad 3& \qquad 4\\ \qquad 4& \qquad -2\end{array}\text{ }|\text{ }\begin{array}{r}\qquad 7\\ \qquad 5\end{array}\right]
We can also write a matrix containing just the coefficients. This is called the coefficient matrix.

[3442]\left[\begin{array}{cc}3& 4\\ 4& -2\end{array}\right]
A three-by-three system of equations such as

3xyz=0 x+y=5 2x3z=2\begin{array}{l}3x-y-z=0\qquad \\ \text{ }x+y=5\qquad \\ \text{ }2x - 3z=2\qquad \end{array}
has a coefficient matrix

[311110203]\left[\begin{array}{rrr}\qquad 3& \qquad -1& \qquad -1\\ \qquad 1& \qquad 1& \qquad 0\\ \qquad 2& \qquad 0& \qquad -3\end{array}\right]
and is represented by the augmented matrix

[311110203  052]\left[\begin{array}{rrr}\qquad 3& \qquad -1& \qquad -1\\ \qquad 1& \qquad 1& \qquad 0\\ \qquad 2& \qquad 0& \qquad -3\end{array}\text{ }|\text{ }\begin{array}{r}\qquad 0\\ \qquad 5\\ \qquad 2\end{array}\right]
Notice that the matrix is written so that the variables line up in their own columns: x-terms go in the first column, y-terms in the second column, and z-terms in the third column. It is very important that each equation is written in standard form
ax+by+cz=dax+by+cz=d
so that the variables line up. When there is a missing variable term in an equation, the coefficient is 0.

How To: Given a system of equations, write an augmented matrix.

  1. Write the coefficients of the x-terms as the numbers down the first column.
  2. Write the coefficients of the y-terms as the numbers down the second column.
  3. If there are z-terms, write the coefficients as the numbers down the third column.
  4. Draw a vertical line and write the constants to the right of the line.


Example: Writing the Augmented Matrix for a System of Equations

Write the augmented matrix for the given system of equations.

 x+2yz=3 2xy+2z=6 x3y+3z=4\begin{array}{l}\text{ }x+2y-z=3\qquad \\ \text{ }2x-y+2z=6\qquad \\ \text{ }x - 3y+3z=4\qquad \end{array}


Try It

Write the augmented matrix of the given system of equations.

4x3y=113x+2y=4\begin{array}{l}4x - 3y=11\\ 3x+2y=4\end{array}


Writing a System of Equations from an Augmented Matrix

We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables. However, it is important to understand how to move back and forth between formats in order to make finding solutions smoother and more intuitive. Here, we will use the information in an augmented matrix to write the system of equations in standard form.

Example: Writing a System of Equations from an Augmented Matrix Form

Find the system of equations from the augmented matrix.

[135254354  256]\left[\begin{array}{rrr}\qquad 1& \qquad -3& \qquad -5\\ \qquad 2& \qquad -5& \qquad -4\\ \qquad -3& \qquad 5& \qquad 4\end{array}\text{ }|\text{ }\begin{array}{r}\qquad -2\\ \qquad 5\\ \qquad 6\end{array}\right]


Try It

Write the system of equations from the augmented matrix.

[111213011519]\left[\begin{array}{ccc}1& -1& 1\\ 2& -1& 3\\ 0& 1& 1\end{array}|\begin{array}{c}5\\ 1\\ -9\end{array}\right]


Row Operations

Now that we can write systems of equations in augmented matrix form, we will examine the various row operations that can be performed on a matrix, such as addition, multiplication by a constant, and interchanging rows.

Performing row operations on a matrix is the method we use for solving a system of equations. In order to solve the system of equations, we want to convert the matrix to row-echelon form, in which there are ones down the main diagonal from the upper left corner to the lower right corner, and zeros in every position below the main diagonal as shown.

Row-echelon form[1ab01d001]\begin{array}{c}\text{Row-echelon form}\\ \left[\begin{array}{ccc}1& a& b\\ 0& 1& d\\ 0& 0& 1\end{array}\right]\end{array}
We use row operations corresponding to equation operations to obtain a new matrix that is row-equivalent in a simpler form. Here are the guidelines to obtaining row-echelon form.

  1. In any nonzero row, the first nonzero number is a 1. It is called a leading 1.
  2. Any all-zero rows are placed at the bottom on the matrix.
  3. Any leading 1 is below and to the right of a previous leading 1.
  4. Any column containing a leading 1 has zeros in all other positions in the column.


To solve a system of equations we can perform the following row operations to convert the coefficient matrix to row-echelon form and do back-substitution to find the solution.

  1. Interchange rows. (Notation:
    RiRj{R}_{i}\leftrightarrow {R}_{j}
    )
  2. Multiply a row by a constant. (Notation:
    cRic{R}_{i}
    )
  3. Add the product of a row multiplied by a constant to another row. (Notation:
    Ri+cRj{R}_{i}+c{R}_{j}
    )


Each of the row operations corresponds to the operations we have already learned to solve systems of equations in three variables. With these operations, there are some key moves that will quickly achieve the goal of writing a matrix in row-echelon form. To obtain a matrix in row-echelon form for finding solutions, we use Gaussian elimination, a method that uses row operations to obtain a 1 as the first entry so that row 1 can be used to convert the remaining rows.

A General Note: Gaussian Elimination

The Gaussian elimination method refers to a strategy used to obtain the row-echelon form of a matrix. The goal is to write matrix
AA
with the number 1 as the entry down the main diagonal and have all zeros below.

A=[a11a12a13a21a22a23a31a32a33]After Gaussian eliminationA=[1b12b1301b23001]A=\left[\begin{array}{rrr}\qquad {a}_{11}& \qquad {a}_{12}& \qquad {a}_{13}\\ \qquad {a}_{21}& \qquad {a}_{22}& \qquad {a}_{23}\\ \qquad {a}_{31}& \qquad {a}_{32}& \qquad {a}_{33}\end{array}\right]\stackrel{\text{After Gaussian elimination}}{\to }A=\left[\begin{array}{rrr}\qquad 1& \qquad {b}_{12}& \qquad {b}_{13}\\ \qquad 0& \qquad 1& \qquad {b}_{23}\\ \qquad 0& \qquad 0& \qquad 1\end{array}\right]
The first step of the Gaussian strategy includes obtaining a 1 as the first entry, so that row 1 may be used to alter the rows below.

How To: Given an augmented matrix, perform row operations to achieve row-echelon form.

  1. The first equation should have a leading coefficient of 1. Interchange rows or multiply by a constant, if necessary.
  2. Use row operations to obtain zeros down the first column below the first entry of 1.
  3. Use row operations to obtain a 1 in row 2, column 2.
  4. Use row operations to obtain zeros down column 2, below the entry of 1.
  5. Use row operations to obtain a 1 in row 3, column 3.
  6. Continue this process for all rows until there is a 1 in every entry down the main diagonal and there are only zeros below.
  7. If any rows contain all zeros, place them at the bottom.


Example: Performing Row Operations on a 3×3 Augmented Matrix to Obtain Row-Echelon Form

Perform row operations on the given matrix to obtain row-echelon form.

[134256334  366]\left[\begin{array}{rrr}\qquad 1& \qquad -3& \qquad 4\\ \qquad 2& \qquad -5& \qquad 6\\ \qquad -3& \qquad 3& \qquad 4\end{array}\text{ }|\text{ }\begin{array}{r}\qquad 3\\ \qquad 6\\ \qquad 6\end{array}\right]




Try It

Write the system of equations in row-echelon form.

 x2y+3z=9 x+3y=42x5y+5z=17\begin{array}{l}\text{ }x - 2y+3z=9\qquad \\ \text{ }-x+3y=-4\qquad \\ 2x - 5y+5z=17\qquad \end{array}


Solve a System With Gaussian Elimination

We have seen how to write a system of equations with an augmented matrix, and then how to use row operations and back-substitution to obtain row-echelon form. Now we will use Gaussian Elimination as a tool for solving a system written as an augmented matrix. In our first example, we will show you the process for using Gaussian Elimination on a system of two equations in two variables.

Example: Solving a 2 X 2 System by Gaussian Elimination

Solve the given system by Gaussian elimination.

2x+3y=6 xy=12\begin{array}{l}2x+3y=6\qquad \\ \text{ }x-y=\frac{1}{2}\qquad \end{array}




Try It

Solve the given system by Gaussian elimination.

4x+3y=11 x3y=1\begin{array}{l}4x+3y=11\qquad \\ \text{ }\text{}\text{}x - 3y=-1\qquad \end{array}


In our next example, we will solve a system of two equations in two variables that is dependent.  Recall that a dependent system has an infinite number of solutions and the result of row operations on it's augmented matrix will be an equation such as
0=00=0
.  We also review writing the general solution to a dependent system.

Example: Solving a Dependent System

Solve the system of equations.

3x+4y=126x+8y=24\begin{array}{l}3x+4y=12\\ 6x+8y=24\end{array}




Now, we will take row-echelon form a step farther to solve a 3 by 3 system of linear equations. The general idea is to eliminate all but one variable using row operations and then back-substitute to solve for the other variables.

Example: Solving a System of Linear Equations Using Matrices

Solve the system of linear equations using matrices.

xy+z=82x+3yz=23x2y9z=9\begin{array}{c}\begin{array}{l}\qquad \\ \qquad \\ x-y+z=8\qquad \end{array}\\ 2x+3y-z=-2\\ 3x - 2y - 9z=9\end{array}





Recall that there are three possible outcomes for solutions to linear systems.  In the previous example, the solution 
(4,3,1)\left(4,-3,1\right)
represents a point in three dimensional space. This point represents the intersection of three planes.  In the next example, we solve a system using row operations and find that it represents a dependent system.  A dependent system in 3 dimensions can be represented by two planes that are identical, much like in 2 dimensions where a dependent system represents two lines that are identical.

Example: Solving a 3 x 3 Dependent System

Solve the following system of linear equations using Gaussian Elimination.

x2y+z=12x+3y=2y2z=0\begin{array}{r}\qquad -x - 2y+z=-1\\ \qquad 2x+3y=2\\ \qquad y - 2z=0\end{array}





The General Solution to a Dependent 3 X 3 System

Recall that when you solve a dependent system of linear equations in two variables using elimination or substitution, you can write the solution
(x,y)(x,y)
in terms of x, because there are infinitely many (x,y) pairs that will satisfy a dependent system of equations, and they all fall on the line 
(x,mx+b)(x, mx+b)
. Now that you are working in three dimensions, the solution will represent a plane, so you would write it in the general form
(x,m1x+b1,m2x+b2)(x, m_{1}x+b_{1}, m_{2}x+b_{2})
.

Try It

Solve the system using Gaussian Elimination.

x+4yz=42x+5y+8z=15x+3y3z=1\begin{array}{c}x+4y-z=4\\ 2x+5y+8z=15\\ x+3y - 3z=1\end{array}





Q & A

Can any system of linear equations be solved by Gaussian elimination?

Yes, a system of linear equations of any size can be solved by Gaussian elimination.

How To: Given a system of equations, solve with matrices using a calculator.

  1. Save the augmented matrix as a matrix variable
    [A],[B],[C],\left[A\right],\left[B\right],\left[C\right]\text{,} \dots
    .
  2. Use the ref( function in the calculator, calling up each matrix variable as needed.


Example: Solving Systems of Equations Using a Calculator

Solve the system of equations.

5x+3y+9z=12x+3yz=2x4y+5z=1\begin{array}{r}\qquad 5x+3y+9z=-1\\ \qquad -2x+3y-z=-2\\ \qquad -x - 4y+5z=1\end{array}





Applications of Systems of Equations

Now we will turn to the applications for which systems of equations are used. In the next example we determine how much money was invested at two different rates given the sum of the interest earned by both accounts.

Example: Applying 2 × 2 Matrices to Finance

Carolyn invests a total of $12,000 in two municipal bonds, one paying 10.5% interest and the other paying 12% interest. The annual interest earned on the two investments last year was $1,335. How much was invested at each rate?





 

Example: Applying 3 × 3 Matrices to Finance

Ava invests a total of $10,000 in three accounts, one paying 5% interest, another paying 8% interest, and the third paying 9% interest. The annual interest earned on the three investments last year was $770. The amount invested at 9% was twice the amount invested at 5%. How much was invested at each rate?





Try It

A small shoe company took out a loan of $1,500,000 to expand their inventory. Part of the money was borrowed at 7%, part was borrowed at 8%, and part was borrowed at 10%. The amount borrowed at 10% was four times the amount borrowed at 7%, and the annual interest on all three loans was $130,500. Use matrices to find the amount borrowed at each rate.





Key Concepts

  • An augmented matrix is one that contains the coefficients and constants of a system of equations.
  • A matrix augmented with the constant column can be represented as the original system of equations.
  • Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows.
  • We can use Gaussian elimination to solve a system of equations.
  • Row operations are performed on matrices to obtain row-echelon form.
  • To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon form. Back-substitute to find the solutions.
  • A calculator can be used to solve systems of equations using matrices.
  • Many real-world problems can be solved using augmented matrices.


Glossary

augmented matrix a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets

coefficient matrix a matrix that contains only the coefficients from a system of equations

Gaussian elimination using elementary row operations to obtain a matrix in row-echelon form

main diagonal entries from the upper left corner diagonally to the lower right corner of a square matrix

row-echelon form after performing row operations, the matrix form that contains ones down the main diagonal and zeros at every space below the diagonal

row-equivalent two matrices
AA
and
BB
are row-equivalent if one can be obtained from the other by performing basic row operations

row operations adding one row to another row, multiplying a row by a constant, interchanging rows, and so on, with the goal of achieving row-echelon form

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