Question

In each case determine how many solutions (and how many parameters) are possible for a homogeneous system of four linear equations in six variables with augmented matrix \(A\). Assume that \(A\) has nonzero entries. Give all possibilities. a. \(\operatorname{Rank} A=2\). b. Rank \(A=1\). c. \(A\) has a row of zeros. d. The row-echelon form of \(A\) has a row of zeros.

Short answer

For a, infinite solutions with 4 parameters; for b, infinite with 5 parameters; for c and d, rank and solutions depend on non-zero rows but always infinite.

Step-by-step solution

Homogeneous System Description

In a homogeneous system of linear equations, of the form \( Ax = 0 \), having more variables than equations implies that this system must have either one or infinitely many solutions. The zero solution (trivial solution) is always one solution, and non-trivial solutions exist if the number of variables exceeds the rank.

Case a - Evaluate Rank 2

For \( \text{Rank } A = 2 \), there are 6 variables and 2 independent equations. The number of free variables will be \( 6 - 2 = 4 \). Therefore, there are infinitely many solutions depending on 4 parameters.

Case b - Evaluate Rank 1

For \( \text{Rank } A = 1 \), we have 6 variables and only 1 independent equation. The number of free variables will be \( 6 - 1 = 5 \). Therefore, there are infinitely many solutions depending on 5 parameters.

Case c - Row of Zeros in A

If \( A \) has a row of zeros, it may decrease the effective number of equations; however, given the context, it simply indicates redundancy in the original set of four equations. Without specific information on non-zero rows, the rank could range from 0 to 3. Since \( A \) has non-zero entries, the minimum rank can be considered as at least 1. Solutions depend on the specific rank as analyzed in prior cases.

Case d - Row-Echelon Form Row of Zeros

If the row-echelon form of \( A \) has a row of zeros, it implies redundancy in the system of equations. This situation is similar to Case c. Here, the row of zeros reduces the rank, so depending on non-zero rows in the echelon form, the possible ranks are between 0 to 3. As explained in the cases where rank is 1 or 2, solutions will depend on the effective non-zero rank.

Key concepts

Augmented Matrix

An augmented matrix is a helpful tool in solving systems of linear equations. It combines the coefficients of the variables with the constants on the right side of the equations, if there are any. This organization allows us to use matrix methods, like row operations, to simplify the system and find solutions.
In the case of a homogeneous system, the augmented matrix includes only the coefficients since the equations are set equal to zero, represented as \( Ax = 0 \).
The augmented matrix helps visualize how equations in the system relate to each other, and its structure can reveal useful information about the system, such as redundancy or dependency among equations.

Rank of a Matrix

The rank of a matrix is the number of linearly independent rows or columns in a matrix. This concept is crucial in understanding the solvability of systems of linear equations.
In a homogeneous system, the rank indicates the number of independent equations present. When compared to the number of variables, the rank helps us determine whether solutions exist beyond the trivial zero solution.
For instance, if the rank of matrix \( A \) is smaller than the number of variables in the system, it implies the presence of free variables, thus suggesting the existence of non-trivial solutions.

Free Variables

Free variables are those that are not leading variables in the row-echelon form of a matrix. In other words, they are the variables that do not appear in the pivot positions when the matrix is simplified. Knowing which variables are free lets us express the solution to the system in terms of these variables.
In a homogeneous system with more variables than independent equations, some variables are inherently free, allowing the construction of solutions that are based on these variables.
This means when the rank is less than the number of variables, the system has infinitely many solutions. Each solution can be described using a set of parameters corresponding to the number of free variables.

Non-trivial Solutions

Non-trivial solutions of a system are any solutions other than the trivial one, where all variables are zero. In the context of a homogeneous system of equations, having non-trivial solutions indicates that not all variables need to be zero to satisfy the system.
Such solutions occur when there are more variables than the rank of the matrix, creating free variables. These free variables can take on various values, leading to a set of solutions that differ from the trivial solution.
This allows us to have a richer set of solutions depending on the specific nature of the system, providing insight into the relationships among the variables in the equations.