Exercises: Rouché-Fröbenius theorem

Two courier companies analyze their routes on a map. Their trajectories are defined by the following equations in a two-dimensional system:

$\begin{cases} x + 2y = 4 \\ 3x + 6y = 12 \end{cases}$

From the graphical representation shown, state the rank of the coefficient matrix $A$ and of the augmented matrix $A^*$, and classify the system using the Rouché-Frobenius theorem.

Given the following system of equations already in row echelon form after applying Gaussian elimination:

$\begin{cases} x - y + 3z = 5 \\ \quad \quad 2y + z = 1 \\ \quad \quad \quad \quad 0z = 0 \end{cases}$

Determine, without solving it, the values of $rg(A)$ and $rg(A^*)$. Is the system consistent determined or indeterminate? Justify your answer according to the number of unknowns.

A financial analyst uses a model with three variables (savings, investment and debt). After calculating the ranks of the matrices associated with the system, he obtains that:

• $rg(A) = 2$

• $rg(A^*) = 3$

Explain with reasons, based on the Rouché-Frobenius theorem, whether the analyst will be able to find a combination of values that satisfies all the conditions of the model.

Consider a homogeneous system of three equations with three unknowns. We know that the determinant of its coefficient matrix is $|A| = 0$.

1. a) What is the value of $rg(A^*)$ compared to $rg(A)$ in any homogeneous system?
2. b) According to the Rouché-Frobenius theorem, can this system be inconsistent? Explain why.

In a video game shop, the prices of three products $x$, $y$, $z$ are governed by the following system of equations:

$\begin{cases} x + y + z = 60 \\ 2x - y + z = 20 \\ 3x + 2z = 80 \end{cases}$

Observe the relationship between the rows of the augmented matrix. If the third equation is the sum of the first two, what is the maximum rank that the coefficient matrix can have? Classify the system.

Given the matrices of a system of linear equations with unknowns $x$ and $y$:

$A = \begin{pmatrix} 1 & 1 \\ 1 & k \end{pmatrix} \quad A^* = \begin{pmatrix} 1 & 1 & 2 \\ 1 & k & 2 \end{pmatrix}$

Determine for which value of $k$ the rank of matrix $A$ is equal to 1. For that value of $k$, what happens to the rank of $A^*$? Classify the resulting system.

Consider a system of three linear equations with three unknowns. Each equation represents a plane in space (three dimensions). We call $A$ the coefficient matrix and $A^*$ the augmented matrix.

Match each of the following rank situations with its geometric interpretation:

1. $rg(A) = rg(A^*) = 3$

2. $rg(A) = rg(A^*) = 2$

3. $rg(A) = 2$ y $rg(A^*) = 3$

Geometry options:

A) The three planes have a line in common (all of them contain it).

B) The three planes intersect at a single point.

C) The three planes have no point in common (for example, they intersect pairwise forming a "prism", or some planes are parallel).

Write the pairing of each number with its letter and briefly justify each choice using the Rouché-Frobenius theorem.

State whether the following statements are True or False, briefly justifying your answer using the Rouché-Frobenius theorem:

1. a) If a system has more equations than unknowns, it is always inconsistent.
2. b) A consistent determined system always satisfies $|A| \neq 0$ ($A$ being the square coefficient matrix).
3. c) If $rg(A) = rg(A^*)$, the system always has a solution.

Marcos is analyzing the consistency of the following system of three equations with three unknowns using the Rouché-Frobenius Theorem:

$$\begin{cases} x + 2y - z = 4 \\ 2x + 4y - 2z = 8 \\ x + 2y - z = 9 \end{cases}$$

After looking at the rows, Marcos states: "Since the second equation is proportional to the first and the third has exactly the same coefficients for the unknowns, the coefficient matrix only has one independent row: $rg(A) = 1$. Since the number of unknowns is $n = 3$ and it holds that $rg(A) < n$, the system is an Underdetermined Consistent System (SCI) and its solution depends on $3 - 1 = 2$ free parameters".

1. a) Identify the basic conceptual error Marcos makes when applying the Rouché-Frobenius Theorem without having analyzed the augmented matrix $A^*$.
2. b) Write the matrices $A$ and $A^*$, and compute their respective ranks ($rg(A)$ and $rg(A^*)$) with reasoned justification.
3. c) Correctly classify the system and indicate which direct contradiction between the equations confirms your conclusion without having to resort to matrix calculations.

Consider the following linear system of two equations with three unknowns depending on a real parameter $k$:

$$\begin{cases} 2x - y + 3z = 5 \\ 4x - 2y + 6z = k \end{cases}$$

1. a) Before carrying out any matrix calculations or row reduction, estimate and answer with reasoned justification: could this system ever be a Determined Consistent System (SCD) for some value of $k$? Justify your prediction by comparing the number of equations ($m = 2$), the number of unknowns ($n = 3$), and the maximum rank that the coefficient matrix $A$ can attain.
2. b) Write the coefficient matrix $A$ and the augmented matrix $A^*$, and compute their ranks by distinguishing the possible cases according to the value of $k$.
3. c) Check whether your initial estimate was correct and formally classify the system in each case by applying the Rouché-Frobenius Theorem.

Consider the following system of equations depending on the real parameter $k$:

$\begin{cases} kx + y + z = 1 \\ x + ky + z = k \\ x + y + kz = k^2 \end{cases}$

Use the Rouché-Frobenius theorem to discuss the consistency of the system according to the different values of $k$. It is not necessary to solve it.

A courier company uses three types of vehicles: vans, light trucks and heavy trucks. The total number of vehicles is 25. It is known that the number of vans is double that of heavy trucks and that, due to a budget restriction, the monthly maintenance cost follows this relationship: $x + my + z = 100$, where $x, y, z$ are the units of each vehicle and $m$ is a variable cost factor.

1. a) Write the system of equations representing the situation.
2. b) Discuss for which values of $m$ the system has a unique solution (that is, when we can know exactly how many vehicles of each type there are).

Given the system of equations:

$\begin{cases} x - 2y + z = 3 \\ 2x + y - 2z = 1 \\ 3x - y - z = a \end{cases}$

Determine, applying the Rouché-Frobenius theorem, the value of parameter $a$ so that the system is consistent indeterminate. Interpret this result in terms of the relative position of the three planes representing the equations.

Study the consistency of the following system of 4 equations with 3 unknowns:

$\begin{cases} x + y + z = 2 \\ 2x + 3y + z = 3 \\ 3x + y + 2z = 5 \\ x - y + z = 1 \end{cases}$

Is it possible to remove one of the equations without changing the solution of the system? Justify your answer by comparing the ranks of the coefficient matrix and the augmented matrix.

An investor divides $10.000$ € between three financial products $A$, $B$ and $C$. The return on product $A$ is $5\%$, that of $B$ is $10\%$ and that of $C$ is $2\%$. At the end of the year, the total profit is $700$ €.

1. a) Prove, using the Rouché-Frobenius theorem, that the system set up is consistent indeterminate.
2. b) If the investor decides that the amount invested in $A$ is equal to the sum of the amounts in $B$ and $C$, does the system become consistent determined? Explain why based on ranks.

Determine the value of $m$ such that the following system is inconsistent:

$\begin{cases} 2x + y - z = 3 \\ x + 2y + z = 2 \\ x - y + mz = 0 \end{cases}$

Explain the meaning of an "inconsistent system" in this context using the concept of rank.

Let the system be:

$\begin{cases} x + y + z = a \\ x + ay + z = 1 \\ ax + y + z = 1 \end{cases}$

1. a) Discuss the system depending on the values of $a$.
2. b) In the cases where the system is consistent, indicate whether the solution will be unique or whether there will be infinitely many solutions based on the number of unknowns and the rank achieved.

Design a system of three linear equations with three unknowns ($x, y, z$) depending on a real parameter $m$ that simultaneously satisfies the following specifications:

1. For any $m \ne 4$, the system must be a Determined Consistent System (SCD).

2. For $m = 4$, the system must be Inconsistent (SI).

To construct it and justify it in a reverse manner:

1. a) Propose a $3 \times 3$ coefficient matrix $A$ whose determinant is $|A| = m - 4$ (or a non-zero multiple of this factor), thus ensuring that $rg(A) = 3$ if and only if $m \ne 4$. Show the calculation of its determinant.
2. b) Choose the constant terms of the system such that, for $m = 4$, it holds that $rg(A^*) = 3$ while $rg(A) = 2$, proving that the system turns out to be Inconsistent according to Rouché-Frobenius.
3. c) What specific modification should you make solely to the constant terms so that, keeping that same matrix $A$ and with $m = 4$, the system becomes an Underdetermined Consistent System (SCI)? Justify it by comparing the resulting ranks.

Given the following system of linear equations depending on the parameter $a \in \mathbb{R}$:

$$\begin{cases} x + y + z = 1 \\ x + a y + z = 2 \\ x + y + a z = a \end{cases}$$

To discuss its consistency according to the values of $a$, two different approaches are proposed:

• Method 1 (Determinants and minors): Compute $|A|$, solve the equation $|A| = 0$ to find the singular values of $a$, and, for each one, border minors to find $rg(A)$ and $rg(A^*)$.

• Method 2 (Gaussian elimination): Apply elementary row operations directly on the augmented matrix $(A|B)$ by subtracting the first row from the others.

1. a) Compare both methods and justify which one is more agile and reliable in this specific case, paying attention to whether Gaussian elimination forces you to divide by expressions containing the parameter $a$.
2. b) Choose one of the two methods, use it to classify the system for all real values of $a$ according to the Rouché-Frobenius Theorem, and state the degrees of freedom in the case of underdetermination.

A high-performance computer assembly company offers three liquid cooling configurations: Basic, Pro, and Extreme.

• The Basic model uses 2 fans, 1 pump, and 3 fittings.

• The Pro model uses 5 fans, 2 pumps, and 8 fittings.

• The Extreme model uses 6 fans, 3 pumps, and $m$ fittings.

It is known that the workshop currently has a stock of 26 fans, 10 pumps, and 42 fittings.

1. a) Set up a system of equations relating the number of computers of each type that can be assembled $(x, y, z)$ using up exactly the stock of fans, pumps, and fittings.
2. b) Use the Rouché-Frobenius Theorem to determine for which value of $m$ the system is consistent undetermined (underdetermined).
3. c) If $m$ is set so that the system is consistent undetermined, is it possible to assemble any computer of the Extreme model if you want to use exactly 42 fittings? Justify your answer.

Determine the values of the real parameters $k$ and $t$ so that the following system is consistent with infinitely many solutions with two degrees of freedom (that is, such that the solution depends on two parameters):

$$\begin{cases} x - 2y + z = t \\ 2x - 4y + 2z = 4 \\ -x + 2y - z = k \end{cases}$$

Explain in detail the relationship between the ranks of the matrix of coefficients and the augmented matrix in this case.

Analyse the consistency of the following system of 4 equations with 3 unknowns depending on the parameter $m$:

$$\begin{cases} x + y + z = 2 \\ 2x + 3y + z = 3 \\ mx + 10y + 4z = 11 \\ x - y + 2z = 3 \end{cases}$$

Is there any value of $m$ for which the system has infinitely many solutions? What about for it to be inconsistent? Base your answer on the rank of the matrices.

Consider the three planes in space whose equations are:

$\pi_1: kx + y + z = 1$

$\pi_2: x + ky + z = 1$

$\pi_3: x + y + kz = 1$

Study their relative position according to the values of $k$.

An investor distributes €100 000 among three types of financial products: government bonds at 3%, tech stocks at 5%, and cryptoassets at 10%.

At the end of the year, the total profits were €6 000. The investor observes that if the returns had been 4% in bonds, 6% in stocks, and 11% in cryptoassets, their total profits would have been $B$ euros.

1. a) Set up the system of equations associated with the problem (calling $x, y, z$ the amounts invested in each product).
2. b) Discuss the consistency of the system depending on the value of $B$.
3. c) Determine the value of $B$ that makes the system Consistent Underdetermined (infinitely many solutions) and explain what this situation means for the investor.

Consider the coefficient matrix $A$ and the matrix of constant terms $B$ of a system of equations:

$A = \begin{pmatrix} 1 & 2 & -1 \\ 0 & 1 & k \\ -1 & m & 2 \end{pmatrix}$, $B = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$

1. a) Determine the values of $m$ and $k$ so that the rank of matrix $A$ is 2.
2. b) For the values found in a), use the Rouché-Frobenius theorem to discuss whether the system $A \cdot X = B$ is consistent or inconsistent.
3. c) If it is inconsistent, what minimal change would you make to matrix $B$ so that the system becomes consistent indeterminate? Justify your answer.

An agricultural cooperative in Almería prepares three types of organic vegetable batches for export: Batch A ($x$), Batch B ($y$), and Batch C ($z$). Each batch requires packaging hours, refrigerated transport volume (in $\text{m}^3$), and generates a $\text{CO}_2$ emission allowance (in kg), leading to the following model:

$$\begin{cases} x + 2y + z = 120 \quad (\text{packaging hours}) \\ 2x + y + 3z = 150 \quad (\text{refrigerated transport}) \\ 3x + 3y + 4z = C \quad (\text{CO}_2 \text{ emission cap}) \end{cases}$$

The regional environmental regulation is currently reviewing the cap $C$, which is a positive parameter.

1. a) Prove using the Rouché-Frobenius Theorem that the viability of the production plan depends exclusively on the value of $C$. What exact value must $C$ have for the system to be consistent? Explain the operational consequence for the cooperative if the administration enforces any value $C \ne 270$.
2. b) For $C = 270$, classify the system and interpret the meaning of the degree of indeterminacy in the cooperative's decision-making.
3. c) Knowing that the number of batches must be non-negative integers ($x, y, z \in \mathbb{N} \cup \{0\}$), find the range of viable values for $z$ and determine how many different production combinations the company can adopt.

To plan the resources of a municipal department, a system of four equations with three unknowns ($x, y, z$) is formulated, involving a budget item $k \in \mathbb{R}$:

$$\begin{cases} x + y + z = 4 \\ 2x - y + z = 1 \\ x + 2y - z = 5 \\ 4x + 2y + z = k \end{cases}$$

Elena, a 2nd year Bachillerato student, argues the following:

"I take the minor of order 3 formed by the first three rows of the coefficient matrix $A$:

$$\begin{vmatrix} 1 & 1 & 1 \\ 2 & -1 & 1 \\ 1 & 2 & -1 \end{vmatrix} = 7 \ne 0$$

Since this minor is non-zero, $rg(A) = 3$. Since there are $n = 3$ unknowns, it holds that $rg(A) = n = 3$. Consequently, by the Rouché-Frobenius Theorem, the system is a Determined Consistent System with a unique solution for any value of $k$, so the fourth equation does not affect the consistency at all".

1. a) Explain in detail what Elena's conceptual error is and why the condition $rg(A) = n$ does not guarantee consistency when there are more equations than unknowns ($m > n$).
2. b) Apply Gaussian elimination to the augmented matrix $A^*$ to row-reduce it and determine the rank of $A^*$ as a function of $k$ with reasoned justification.
3. c) State the correct discussion of the system according to the Rouché-Frobenius Theorem, specifying for which value of $k$ a solution exists and what happens if $k$ takes any other value.