Problem 55 - 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. $$ \begin{aligned} x_{1}+x_{2}-x_{3}+x_{4} &=0 \\ 5 x_{2}+2 x_{4} &=0 \\ x_{1}+x_{3}-x_{4} &=0 \\ 3 x_{1}+2 x_{2}-x_{3}+x_{4} &=0 \end{aligned} $$

Step-by-Step Solution

Answer

The system has a nontrivial solution.

Step 1: Identify the system

We have a homogeneous system of linear equations. The given system can be rewritten as a matrix equation of the form \( A\mathbf{x} = \mathbf{0} \), where \( A \) is the coefficient matrix and \( \mathbf{x} \) is the vector of variables. For this problem, \( A \) is a 4x4 matrix:

Step 2: Write the coefficient matrix

The matrix \( A \) corresponding to the given system is:\[A = \begin{bmatrix}1 & 1 & -1 & 1 \0 & 5 & 0 & 2 \1 & 0 & 1 & -1 \3 & 2 & -1 & 1\end{bmatrix}\]

Step 3: Determine the rank of the matrix

The next step is to determine the rank of the matrix \( A \). The rank can be found by transforming the matrix into row-echelon form or reduced row-echelon form.

Step 4: Analyze rows in reduced form

Perform row operations to simplify \( A \), observing which rows can be reduced to zero. Count the number of non-zero rows to determine the rank. This matrix, upon row reducing, has 3 non-zero rows indicating the rank is 3.

Step 5: Compare rank to number of variables

Compare the rank of the matrix, which is 3, with the number of variables \( n = 4 \). Since the rank is less than the number of variables, the system has infinitely many solutions, indicating the presence of a nontrivial solution.

Key Concepts

Matrix Equation

A matrix equation is an efficient way to represent a system of linear equations. Consider a set of linear equations, for instance, \(4\) equations involving \(4\) unknowns, as seen in our exercise. Representing it as a matrix equation allows leveraging matrix operations for finding solutions. This is denoted as \( A\mathbf{x} = \mathbf{0} \), where \( A \) is the matrix of coefficients, and \( \mathbf{x} \) is the column vector of unknown variables, representing the solutions to the system.

• In the context of homogeneous systems, \( \mathbf{0} \) signifies the zero vector, indicating all equations equal zero.
• This is often used because it simplifies operations and transforms potentially complex algebraic manipulation into more straightforward matrix operations.

Working with matrices often makes it easier to apply theoretical results from linear algebra, such as row reduction methods and rank calculations, enabling well-defined approaches to understanding the nature of solutions.

Row-Echelon Form

Row-echelon form is a specific arrangement of a matrix that makes solving systems of linear equations simpler and more intuitive. A matrix is said to be in row-echelon form if:

• All non-zero rows (rows with at least one non-zero element) are above any rows of all zeroes.
• Every leading coefficient (the first non-zero number in a row) of a row is to the right of the leading coefficient of the previous row.

The process of converting a matrix to this form involves performing row operations, such as swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of rows from each other. By transforming a matrix into this form, one can easily determine the rank,
which can help in identifying the solutions to the system of equations it represents.Reaching row-echelon form is a precursor to reduced row-echelon form, where further simplification occurs by making all leading coefficients \(1\) and ensuring all other elements in the column containing a leading \(1\) are zeros.

Rank of a Matrix

The rank of a matrix is a fundamental concept in linear algebra. It represents the number of linearly independent rows or columns in the matrix, essentially measuring the dimension of the vector space spanned by its rows or columns.
For the matrix in our problem, transforming it to row-echelon or reduced row-echelon form makes it easier to determine the rank. Once in row-echelon form, the rank is simply the number of non-zero rows.

• If the rank of a matrix equals the number of its columns, the system it represents typically has a unique solution if it was non-homogeneous.
• In the context of homogeneous systems, if the rank is less than the number of variables, it indicates that the system has infinitely many solutions. This implies the existence of nontrivial solutions other than the trivial (zero) solution.

Determining the rank helps assess the solvability of the system and whether it has trivial or nontrivial solutions, which is crucial for understanding the nature of the solutions.