Problem 54 - Advanced Engineering Mathematics

Question

In Problems 53-56, without solving, determine whether the given homogeneous system of equations has only the trivial solution or a nontrivial solution. $$ \begin{array}{r} x_{1}+x_{2}+x_{3}=0 \\ x_{1}-2 x_{2}+x_{3}=0 \\ -2 x_{1}+x_{3}-2 x_{3}=0 \end{array} $$

Step-by-Step Solution

Answer

The system has a nontrivial solution.

Step 1: Understand the System

The given system of equations is homogeneous, meaning the right side of each equation is 0. We need to determine if the system has only the trivial solution (where all variables are zero).

Step 2: Write the System as a Matrix

Construct the coefficient matrix from the given system of equations: \[ A = \begin{bmatrix} 1 & 1 & 1 \ 1 & -2 & 1 \ -2 & 0 & -2 \end{bmatrix} \].

Step 3: Reduce the Matrix to Row Echelon Form

Perform row operations to reduce the matrix to row-echelon form. A step-by-step process:- Swap row 1 with row 2 to move a leading 1 to the top: \[ \begin{bmatrix} 1 & -2 & 1 \ 1 & 1 & 1 \ -2 & 0 & -2 \end{bmatrix} \]- Subtract row 1 from row 2: \[ \begin{bmatrix} 1 & -2 & 1 \ 0 & 3 & 0 \ -2 & 0 & -2 \end{bmatrix} \]- Add 2 times row 1 to row 3: \[ \begin{bmatrix} 1 & -2 & 1 \ 0 & 3 & 0 \ 0 & -4 & 0 \end{bmatrix} \]

Step 4: Analyze the Matrix Rank

The row-echelon form shows that there are pivot positions in at least two of the rows. Pivot columns are defined in columns 1 and 2. To complete the analysis, continue reducing to see if any row becomes a row of zeros, indicating that the system might have free variables.

Step 5: Further Reduce and Determine Independence

Divide row 2 by 3: \[ \begin{bmatrix} 1 & -2 & 1 \ 0 & 1 & 0 \ 0 & -4 & 0 \end{bmatrix} \]. Now continue to eliminate the second column below the pivot:- Add 4 times row 2 to row 3: \[ \begin{bmatrix} 1 & -2 & 1 \ 0 & 1 & 0 \ 0 & 0 & 0 \end{bmatrix} \]. The matrix rank is 2 (there are 2 pivot positions), which matches the number of unknowns minus 1 (3-1=2). Therefore, there is likely a free variable indicating a nontrivial solution.

Key Concepts

Matrix Row Reduction

Matrix row reduction is a valuable process used to simplify a matrix to make solving systems of equations easier. This is typically done by transforming a given matrix to row-echelon form or even further to reduced row-echelon form. Here’s how you can think about it:

• Each row of the matrix represents one equation.
• The goal is to create zeros below the leading "1" in each column to simplify back-substitution.
• The series of operations applied includes swapping rows, multiplying rows by a nonzero constant, and adding or subtracting rows from one another.

In our exercise, row reduction started by swapping rows to place a suitable pivot at the top. This was followed by eliminating other entries in the column to streamline solving. The matrix row reduction pushes equations toward simpler forms, ultimately leading to a decision about the solutions: trivial or nontrivial.

Trivial Solution

The trivial solution in a homogeneous system of equations occurs when all variables equal zero. This solution exists in all homogeneous systems and is always a valid solution. Here's what you need to know:

• In equation form, it's like saying \( x_1 = 0, x_2 = 0, x_3 = 0 \).
• Every homogeneous system of linear equations has at least this solution. It's the basic "do nothing" solution.

In practical terms, if the only solution after matrix row reduction is zero for all variables, it's labeled as the trivial solution. The presence or absence of only the trivial solution often hinges on the row rank of the reduced matrix and whether it matches the number of variables.

Nontrivial Solution

A nontrivial solution refers to any solution of a homogeneous system where at least one variable is not zero. Identifying whether such solutions exist is highly dependent on the matrix's structure after reduction.

• They exist when the number of pivot positions is less than the number of variables.
• This will result in at least one free variable, leading to infinite solutions.

In the exercise, by reducing the matrix into row-echelon form, we see fewer pivot positions than variables (rank < number of columns). This indicates the system has nontrivial solutions, signaling that at least one variable can take on values other than zero. The presence of free variables allows for a set of non-zero values to also satisfy the equations, contributing to complex solution sets essential in many applications like physics and computer graphics.