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Magazine Chapter 8: Problem 20
In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{aligned} x_{1}+2 x_{2}+\quad x_{4} &=0 \\ 4 x_{1}+9 x_{2}+x_{3}+12 x_{4} &=0 \\ 3 x_{1}+9 x_{2}+6 x_{3}+21 x_{4} &=0 \\ x_{1}+3 x_{2}+x_{3}+9 x_{4} &=0 \end{aligned} $$
Short Answer
Expert verified
The system has infinitely many solutions: \(x_1 = 19s\), \(x_2 = -10s\), \(x_3 = 2s\), \(x_4 = s\).
Step by step solution
01
Write the Augmented Matrix
First, express the system of equations as an augmented matrix. For our system: \[\begin{bmatrix}1 & 2 & 0 & 1 & | & 0 \4 & 9 & 1 & 12 & | & 0 \3 & 9 & 6 & 21 & | & 0 \1 & 3 & 1 & 9 & | & 0 \\end{bmatrix}\]
02
Perform Row Operations to Get Leading Ones
Begin Gaussian elimination by creating leading 1s in the first column. We'll use the first row as a pivot and eliminate the elements below it in the first column:- R2 = R2 - 4*R1- R3 = R3 - 3*R1- R4 = R4 - R1After performing these operations, the matrix becomes:\[\begin{bmatrix}1 & 2 & 0 & 1 & | & 0 \0 & 1 & 1 & 8 & | & 0 \0 & 3 & 6 & 18 & | & 0 \0 & 1 & 1 & 8 & | & 0 \\end{bmatrix}\]
03
Simplify the Matrix Further
Continue to simplify starting with the second row. We need to ensure the second column under the pivot in R2 is zero:- R3 = R3 - 3*R2- The matrix then becomes:\[\begin{bmatrix}1 & 2 & 0 & 1 & | & 0 \0 & 1 & 1 & 8 & | & 0 \0 & 0 & 3 & -6 & | & 0 \0 & 1 & 1 & 8 & | & 0 \\end{bmatrix}\]
04
Eliminate Duplicate Rows
Observe that R2 and R4 are identical, meaning we can perform the operation:- R4 = R4 - R2, which reduces R4 to:\[\begin{bmatrix}0 & 0 & 0 & 0 & | & 0\end{bmatrix}\]Our updated matrix now looks like:\[\begin{bmatrix}1 & 2 & 0 & 1 & | & 0 \0 & 1 & 1 & 8 & | & 0 \0 & 0 & 3 & -6 & | & 0 \0 & 0 & 0 & 0 & | & 0 \\end{bmatrix}\]
05
Back-Substitution and Solutions
Since our final row reduced to zeros, and there are no contradictions, the system has solutions. Notice that we have two free variables due to the rank of this system being 2 while having 4 variables.This means \( x_3 \) and \( x_4 \) can be parameters \( t \) and \( s \) respectively, where:- \( x_4 = s \)- \( x_3 = 2s \)- From the second row: \( x_2 = -x_3 - 8x_4 = -2s - 8s = -10s \)- Substituting these into the first row: \( x_1 = -2(-10s) - 1(s) = 20s - s = 19s \)Thus, \( x_1 = 19s \), \( x_2 = -10s \), \( x_3 = 2s \), and \( x_4 = s \).
06
Conclusion: General Solution
The solutions are given in terms of the parameter \( s \): \ \[ x_1 = 19s, \quad x_2 = -10s, \quad x_3 = 2s, \quad x_4 = s\] Every such vector \([19s, -10s, 2s, s]^T\) is a solution.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Gauss-Jordan Elimination
Gauss-Jordan elimination is a method used to solve systems of linear equations. Unlike Gaussian elimination, which stops at obtaining a row-echelon form, Gauss-Jordan elimination continues to transform the augmented matrix into reduced row-echelon form. This means you aim for a form where zeros appear in all positions above and below each leading 1.
Gauss-Jordan is comprehensive. It leaves you with a clear path to identify solutions, be they unique, infinite, or nonexistent. Here are some steps generally involved in this method:
- Transform each row to make the leading coefficient (first non-zero element) a 1, if it is not already.
- Clear out other elements in the column by subtracting suitable multiples of the pivot row from the others.
- Since it's similar to back-substitution, once in reduced form, variables can be easily solved for.
Augmented Matrix
An augmented matrix is a useful tool in linear algebra that combines two related data sets into a single matrix. Typically, this involves the transformation of a system of linear equations, pairing the coefficient matrix with the constants on the right-hand side.In our exercise, the system of equations is transformed into the following format:\[\begin{bmatrix}1 & 2 & 0 & 1 & | & 0 \4 & 9 & 1 & 12 & | & 0 \3 & 9 & 6 & 21 & | & 0 \1 & 3 & 1 & 9 & | & 0 \\end{bmatrix}\]
- The vertical line distinguishes the coefficients from the constants.
- This matrix is manipulated to solve the system using Gauss-Jordan or other methods.
Row Operations
Row operations are the legal moves you can make on an augmented matrix to simplify solving systems of equations. There are three primary types of row operations:
- Row swapping: swapping two rows to bring a crucial pivot element into position.
- Row multiplication: multiplying a row by a non-zero scalar to simplify the leading coefficient.
- Row addition: Adding or subtracting multiples of one row from another to zero out elements as needed.
Free Variables
Free variables arise when there are fewer leading 1s than the number of variables in a matrix reduced by Gauss-Jordan elimination. These can be adjusted freely as parameters in the solution set.In our exercise, we have the final form identifying two free variables. Since the system involves four variables but only two leading 1s (rank 2), this indicates two variables can take any value. Typically, free variables are set as parameters such as \( t \) or \( s \):
- In the present case: \( x_3 = 2s \) and \( x_4 = s \)
- Solutions can then be expressed in terms of these parameters for other variables: \( x_1 \) and \( x_2 \).
