Question

In each case either show that the statement is true, or give an example \(^{2}\) showing it is false. a. If a linear system has \(n\) variables and \(m\) equations, then the augmented matrix has \(n\) rows. b. A consistent linear system must have infinitely many solutions. c. If a row operation is done to a consistent linear system, the resulting system must be consistent. d. If a series of row operations on a linear system results in an inconsistent system, the original system is inconsistent.

Short answer

a. False; b. False; c. True; d. True.

Step-by-step solution

Understanding the Augmented Matrix

The statement is about augmented matrices. For a system with \(n\) variables, the augmented matrix includes an additional column compared to the coefficient matrix, but the number of rows is determined by the number of equations \(m\). Thus, the augmented matrix has \(m\) rows, not \(n\) rows.

Analyzing Consistency and Solutions

A consistent linear system is one where at least one solution exists. However, consistency does not necessarily mean it has infinitely many solutions; it could also have a unique solution. Therefore, a consistent system can either be unique or infinitely many.

Impact of Row Operations on Consistency

Performing row operations like row swapping, multiplication, or adding multiples of rows affects the form but not the nature of the solution. If a system is initially consistent, it remains consistent after any series of valid row operations.

Relating Final and Original System Consistency

If performing row operations on a consistent system results in inconsistency (such as obtaining a row like [0 0 ... 0 | 1]), it implies no solutions exist. However, this is impossible for a consistent system; hence the original system must have been inconsistent if the result is inconsistent.

Key concepts

Augmented Matrix

An augmented matrix is a helpful tool used to represent a system of linear equations. It combines the coefficient matrix with the constant terms from the equations into one unified matrix.
For example, consider a linear system with equations and variables. The coefficient matrix comprises only the coefficients of the variables, while the augmented matrix includes an additional column for constants. This means:

• The number of rows in the augmented matrix corresponds to the number of equations, \(m\).
• The augmented matrix has one more column than the coefficient matrix, accounting for the constant terms.

The misconception that the number of rows equals the number of variables often arises because students focus on variables without considering equations. Understanding this distinction is crucial in linear algebra exercises involving matrices.

Row Operations

Row operations are crucial techniques used when manipulating matrices, especially when solving linear systems. These operations include:

• Swapping: Interchanging two rows in a matrix.
• Multiplication: Multiplying all entries of a row by a non-zero constant.
• Addition: Replacing a row by the sum of itself and a multiple of another row.

These operations are fundamental in obtaining specific forms of matrices such as row-echelon form and reduced row-echelon form, which simplify the process of solving the linear system.
The beauty of these operations is that they preserve the solution set of the system. Any valid row operations will not alter the consistency of a consistent system. So, if you start with a consistent system, you can be sure it will remain consistent, no matter how many valid row operations you perform.

Consistency of Linear Systems

When dealing with linear systems, consistency is a key concept. A system is called consistent when it has at least one solution. There are two primary scenarios for consistent systems:

• Unique Solution: The system leads to a single solution, meaning the lines intersect at one point.
• Infinitely Many Solutions: The system leads to multiple solutions, usually when the lines coincide entirely.

An inconsistent system has no solution because the equations describe lines or planes that never intersect, like parallel lines in two dimensions. When row operations applied to a system lead to a situation like [0 0 ... 0 | 1], it confirms inconsistency, signifying no possible values for variables.
Therefore, the detection of inconsistency after row operations implies the initial system itself was inconsistent, assuring no solution existed from the start. Recognizing this outcome helps in understanding the behavior of the system post-manipulation.