Problem 14 - Advanced Engineering Mathematics

Question

In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{array}{r} x_{1}-2 x_{2}+x_{3}=2 \\ 3 x_{1}-x_{2}+2 x_{3}=5 \\ 2 x_{1}+x_{2}+x_{3}=1 \end{array} $$

Step-by-Step Solution

Answer

The system is inconsistent; no solution exists.

Step 1: Representing the system as an augmented matrix

Write the system of equations as an augmented matrix: \[\begin{bmatrix}1 & -2 & 1 & | & 2 \3 & -1 & 2 & | & 5 \2 & 1 & 1 & | & 1\end{bmatrix}\]

Step 2: Row operations for upper triangular form

Perform row operations to get an upper triangular form. First, eliminate the first element below the first pivot (the element in the first row, first column) by subtracting appropriate multiples of the first row from subsequent rows:- Replace \(R_2\) with \(R_2 - 3R_1\): \[ \begin{bmatrix} 1 & -2 & 1 & | & 2 \ 0 & 5 & -1 & | & -1 \ 2 & 1 & 1 & | & 1 \end{bmatrix} \]- Replace \(R_3\) with \(R_3 - 2R_1\): \[ \begin{bmatrix} 1 & -2 & 1 & | & 2 \ 0 & 5 & -1 & | & -1 \ 0 & 5 & -1 & | & -3 \end{bmatrix} \]

Step 3: Make diagonal elements of middle row null below the pivot

Subtract \(R_2\) from \(R_3\) to eliminate the element below the second pivot:\[\text{Replace } R_3 \text{ with } R_3 - R_2\]Now the matrix becomes:\[\begin{bmatrix}1 & -2 & 1 & | & 2 \0 & 5 & -1 & | & -1 \0 & 0 & 0 & | & -2\end{bmatrix}\]

Step 4: Check for inconsistency

Observe the last row of the matrix: \([0, 0, 0 | -2]\). This row implies \(0x_1 + 0x_2 + 0x_3 = -2\), which is a contradiction, indicating the system is inconsistent and does not have a solution.

Key Concepts

Understanding Augmented Matrices

An augmented matrix is a way to represent a system of linear equations. It is incredibly useful for solving systems using methods such as Gaussian elimination.
Think of it as a compact version of the system where the coefficients of the variables and the constants from the equations are all in one matrix.
In our example, the system of equations transformed into an augmented matrix looks like this:

• The numbers to the left of the vertical bar are the coefficients from each equation.
• The numbers to the right are the constants.

Using an augmented matrix helps streamline row operations, making it less error-prone during elimination processes.

Role and Execution of Row Operations

Row operations are steps you can perform to manipulate a matrix in order to simplify it or find solutions to the associated system of equations. You can add, subtract, or multiply whole rows by numbers and swap rows around.
This flexibility lets you move towards an upper triangular or reduced row echelon form, which is key in techniques like Gaussian or Gauss-Jordan elimination.
For example, in Step 2 from the original solution, row operations were used to zero out elements below the pivot to help transform the matrix structure.

• Eliminating elements below pivots makes solving for unknowns easier.
• The goal is to simplify the matrix to a point where the solution is either clear, or the matrix form indicates a different status, like inconsistency.

Identifying an Inconsistent System

An inconsistent system in linear algebra means that no solutions satisfy all equations simultaneously.
The inconsistency usually surfaces during Gaussian elimination or similar methods when you arrive at a row in your matrix that implies something impossible, such as saying zero equals non-zero.
In our original example, we concluded the system was inconsistent when we had a row \(\[0, 0, 0 | -2\]\) in the matrix:

• It translates to \(0x_1 + 0x_2 + 0x_3 = -2\), which is not plausible under any circumstance.
• Such a row indicates a contradiction, and thus the system has no solutions.

The detection of inconsistencies is crucial as it tells us the system of equations does not have intersecting solutions in the real number space.