Question

Consider a system of linear equations with augmented matrix \(A\) and coefficient matrix \(C\). In each case either prove the statement or give an example showing that it is false. a. If there is more than one solution, \(A\) has a row of zeros. b. If \(A\) has a row of zeros, there is more than one solution. c. If there is no solution, the reduced row-echelon form of \(C\) has a row of zeros. d. If the row-echelon form of \(C\) has a row of zeros, there is no solution. e. There is no system that is inconsistent for every choice of constants. f. If the system is consistent for some choice of constants, it is consistent for every choice of constants.

Short answer

(a) False; (b) False; (c) False; (d) False; (e) True; (f) False.

Step-by-step solution

Understanding Statements

We need to evaluate each statement regarding the properties of an augmented matrix \(A\) from a system of linear equations and its coefficient matrix \(C\). We'll either prove each statement or provide a counterexample to show it's false.

Evaluating (a) for Multiple Solutions and Row of Zeros

Statement (a) says if there is more than one solution, \(A\) has a row of zeros. Multiple solutions occur in systems with free variables, often when there's a row in the echelon form with zero coefficients and a zero in the augmented column. However, this isn't a necessity as free variables can exist without there being a zero row, making the statement false. A counterexample is the system \(x + y = 1\). Here, the RREF of \(A\) isn't required to have a row of zeros.

Evaluating (b) for Row of Zeros and Multiple Solutions

Statement (b) implies that if \(A\) has a row of zeros, there are multiple solutions. This can be false. A row of zeros could indicate redundancy, not necessarily indicating free variables if other rows still uniquely determine a solution. Consider the system \(x = 1\) and \(0 = 0\) as represented by \([1 \, 0 \, | \, 1; \, 0 \, 0 \, | \, 0]\) - it has a unique solution.

Evaluating (c) for No Solution and Row of Zeros in RREF of C

Statement (c) claims that if there's no solution, the RREF of \(C\) has a row of zeros, which is incorrect. No solution occurs when an impossible equation arises, like a row \([0 \, 0 \, | \, 1]\) in RREF of \(A\), showing inconsistency but no zero row in \(C\). RREF of \(C\) could merely indicate dependency among equations, not the unsolvable nature of the system.

Evaluating (d) for Row of Zeros in RREF of C and No Solution

Statement (d) suggests that if the RREF of \(C\) has a row of zeros, there's no solution. A zero row in RREF of \(C\) indicates free variables and potentially infinite solutions, not necessarily indicating the system has no solution. Example: \(x + y + z = 1\), with RREF of \([1 \, 0 \, | \, 1; \, 0 \, 0 \, | \, 0]\) has infinite solutions.

Evaluating (e) for Systems Inconsistent for Every Constants Choice

Statement (e) argues that there's no system always inconsistent regardless of constants. A consistent system has some solution unless it directly implies contradiction for all constant choices. Consider the system \(x = 1 \land x = x+1\), being flawed and impossible inherently, but typically systems are consistent for suitable constants.

Evaluating (f) for Consistency Across All Constant Choices

Statement (f) claims if consistent for some constants, it’s consistent for all. This is false; consistency depends on specific scenarios. A solvable condition for one constant choice, like \((x = 1)\), could be inconsistent for another, like \((x = 0 \land x = 1)\). Different constants can change solution viability.

Key concepts

Augmented Matrix

An augmented matrix is a representation of a system of linear equations in a simple rectangular array. It contains the coefficient matrix with the additional column representing the constants from each equation. This form allows us to perform row operations and simplify the process of finding solutions.

The augmented matrix typically has the form \[\begin{bmatrix} a_{11} & a_{12} & \ldots & a_{1n} & | & b_1 \\vdots & \vdots & & \vdots & & \vdots \a_{m1} & a_{m2} & \ldots & a_{mn} & | & b_m \end{bmatrix}\]Here, the vertical bar is used to separate the coefficients of the variables from the constants on the right-hand side of the equations.

When solving systems of equations, the augmented matrix allows us to perform manipulations that simultaneously affect all equations in the system. Operations we can perform include:

• Swapping two rows
• Multiplying a row by a non-zero scalar
• Adding or subtracting rows from each other

These operations aim to transform the augmented matrix into simpler forms, such as row echelon form or reduced row echelon form, to easily identify solutions or inconsistencies.

Coefficient Matrix

The coefficient matrix is a crucial element of linear algebra when dealing with systems of linear equations. It includes only the coefficients of the variables, excluding the constant terms. For example, in the system of equations represented by \[\begin{align*}x + y &= 5 \2x - 3y &= 4 \end{align*}\]the coefficient matrix would be:\[\begin{bmatrix}1 & 1 \2 & -3 \end{bmatrix}\]This matrix provides a compact way to organize all the coefficients, making it easier to apply linear transformations and solve the system using tools such as elimination or substitution.

A key aspect to remember is that the coefficient matrix alone does not contain the information about the solution of the system. It only gives us a structure to help manage and work through the variable components of the equations.

Row Echelon Form

Row echelon form (REF) is a simplified version of a matrix, achieved through a series of row operations. In this form, the steps are taken to make matrices easier to handle and solve in terms of finding solutions to systems of linear equations. A matrix is said to be in row echelon form if:

• All non-zero rows are above any rows of all zeroes.
• The leading coefficient (also called the pivot) of a non-zero row is always to the right of the leading coefficient in the row above it.
• All entries in a column below a leading entry are zero.

Here's a simple example:\[\begin{bmatrix}1 & 2 & 3 \0 & 1 & 4 \end{bmatrix}\]Transforming a matrix into row echelon form is often the first step in solving systems by means of Gaussian elimination. Once in REF, we can further simplify into reduced row echelon form (RREF) where each leading entry is 1 and is the only non-zero entry in its column.

Consistency and Inconsistency of Systems

When we talk about the consistency of a system of linear equations, we're addressing whether the system has solutions.

• A consistent system has at least one solution. It could be either one unique solution or infinitely many solutions.
• An inconsistent system has no solutions due to contradictions between equations.

For instance, the system:\[\begin{align*}x + y &= 2 \x + y &= 3 \end{align*}\]is inconsistent because it leads to a logical contradiction (the same expression equals two different constants simultaneously).

To determine consistency, we perform row operations to simplify the augmented matrix, often looking for:

• If a row of form \([0 \, 0 \, | \, c]\) (where \(c eq 0\)) emerges, the system is inconsistent.
• An absence of such rows indicates the system might be consistent, potentially having free variables or parametric solutions.

Understanding consistency is essential for predicting the behavior and solutions of a system of equations.