Exercises: Rouché-Frobenius Theorem

In an algebra study session, a classmate shows you the results of analysing the ranks of three different systems of equations using the Rouché-Frobenius Theorem. State with reasons the type of system (Consistent with a unique solution, Consistent with infinitely many solutions, or Inconsistent) in each of the following cases, knowing that the number of unknowns is $n = 3$:

1. a) $rg(A) = 3$ and $rg(A^*) = 3$.
2. b) $rg(A) = 2$ and $rg(A*) = 3$.
3. c) $rg(A) = 2$ and $rg(A^*) = 2$.

Given the following system of linear equations without parameters:

$$\begin{cases} x + y + z = 2 \\ 2x - y + 3z = 5 \\ 3x + 4z = 7 \end{cases}$$

Calculate the rank of the coefficient matrix $A$ and the rank of the augmented matrix $A^*$. Use the Rouché–Frobenius theorem to discuss the existence and number of solutions of the system.

Discuss the following system of equations as a function of the real parameter $k$:

$$\begin{cases} kx + y = 1 \\ x + y = k \end{cases}$$

Determine for which values of $k$ the system is Consistent and Determinate (unique solution), and for which it is Inconsistent or Consistent and Indeterminate (infinitely many solutions).

Consider a homogeneous system of three equations in three unknowns whose coefficient matrix is $A$. If it is known that the determinant of this matrix is $|A| = 0$ and that there exists a non-zero minor of order 2, use the Rouché–Frobenius theorem to:

1. a) Justify why the system can never be inconsistent.
2. b) Classify the system according to the number of solutions.
3. c) State how many parameters will be needed to express its general solution.

An "influencer" is analysing the growth of their followers across three different social networks (TikTok, Instagram, and Twitch). They have set up a system of equations to distribute their advertising budget, but have realised that there are more unknowns than equations:

$$\begin{cases} x + 2y + z = 1000 \\ 2x - y + 3z = 1500 \end{cases}$$

Where $x, y, z$ is the investment in euros for each social network. Without solving the system, explain with reasons using the Rouché–Frobenius theorem why this system can never have a unique solution. Could it be inconsistent?

Match each of the following geometric configurations of three planes in space with the discussion of the corresponding system according to the Rouché–Frobenius theorem (assume $n=3$):

1. The three planes intersect at a single point.

2. Two planes are parallel and the third intersects them.

3. The three planes intersect along a common line.

Match each case with one of these options:

1. a) $rg(A) = 2, rg(A^*) = 2$.
2. b) $rg(A) = 3, rg(A^*) = 3$.
3. c) $rg(A) = 2, rg(A^*) = 3$.

Study the consistency of the following system of equations depending on the value of the parameter $a$, which only affects the constant term of the third equation:

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

Is there any value of $a$ for which the system is consistent with infinitely many solutions? Justify your answer by calculating the ranks of $A$ and $A^*$.

Discuss the nature of the system formed by the equations of three lines in the plane, knowing that the system is given by:

$$\begin{cases} 2x + 3y = 5 \\ x - y = 1 \\ 4x + 6y = 10 \end{cases}$$

Calculate the rank of the coefficient matrix and of the augmented matrix. Interpret the result obtained: do the three lines intersect at a point, are they the same line, or is there any parallel relationship?

Lucas is solving the following system of 3 linear equations in 3 unknowns:

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

After calculating the determinant of the coefficient matrix and obtaining $|A| = 0$, he states on his exam:

"Since the determinant of the coefficient matrix is zero, the three equations contradict each other and, by the Rouché-Frobenius Theorem, the system is inconsistent, having no solution."

1. a) Identify and explain in a reasoned manner the conceptual error Lucas makes regarding the relationship between a zero determinant and the classification of the system.
2. b) Without solving the system, calculate the rank of the coefficient matrix $A$ and that of the augmented matrix $A^*$, and state the correct conclusion derived from the Rouché-Frobenius Theorem.

Consider the following linear system of 4 equations in 3 unknowns:

$$\begin{cases} x - 2y + z = 1 \\ 2x + y - z = 3 \\ -x + 7y - 4z = 0 \\ 3x - y = 4 \end{cases}$$

1. a) Estimate in a reasoned manner, before performing any matrix calculations:

• What is the theoretical maximum rank that the coefficient matrix $A$ and the augmented matrix $A^*$ can achieve?

• Looking carefully only at the first two equations, would it be possible for the system to be indeterminate consistent with 2 degrees of freedom (that is, with infinitely many solutions depending on two parameters)? Justify your estimation.

1. b) Verify your estimates by calculating the ranks $rg(A)$ and $rg(A^*)$ using the Gaussian elimination method, and classify the system definitively according to the Rouché-Frobenius Theorem.

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

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

Discuss its consistency according to the values of $k$ using the Rouché-Frobenius Theorem. It is not necessary to solve it.

Consider the three planes represented in space whose equations are given below:

$$\pi_1: x + ay + z = 1$$ $$\pi_2: ax + y + (a-1)z = a$$ $$\pi_3: x + y + z = a + 1$$

Study the relative position of the three planes as a function of the parameter $a$ based on the rank of the coefficient matrix $A$ and the augmented matrix $A^*$.

An electronic components company manufactures three types of motherboards: Alpha, Beta and Gamma. It is known that:

• The total number of motherboards manufactured in a day is $150$ units.

• The production cost of each Alpha motherboard is $50$ €, that of Beta is $60$ € and that of Gamma is $80$ €. The total daily cost is $10.000$ €.

• The sum of the Alpha and Beta motherboards manufactured exceeds the number of Gamma motherboards by $m$ units.

Set up the system of equations and discuss, using the Rouché-Frobenius Theorem, for which values of the parameter $m$ the problem has a unique solution.

Consider the system of linear equations:

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

Discuss the system depending on the real parameters $a$ and $b$. Determine under what conditions the system is consistent with infinitely many solutions and provide the geometric interpretation of that case.

Study the consistency of the following homogeneous system as a function of the parameter $m$:

$$\begin{cases} x + y + z = 0 \\ mx + 2y + mz = 0 \\ 2x + my + 4z = 0 \end{cases}$$

State for which values of $m$ the system admits non-trivial solutions $(0, 0, 0)$.

A third equation is added to a system of two equations in three unknowns that is consistent with infinitely many solutions.

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

Find an equation of the form $Ax + By + Cz = D$ such that the resulting system is:

1. Inconsistent.

2. Consistent with a unique solution.

3. Consistent with infinitely many solutions.

Justify your answer by analysing the ranks of the coefficient matrix and the augmented matrix.

Given the coefficient matrix $A$ and the matrix of constant terms $B$ of a system $AX = B$:

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

Use the determinant of $A$ to discuss the system. Is there any value of $m$ for which the rank of the augmented matrix $A^*$ is $3$ while the rank of $A$ is $2$?

Consider the system:

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

1. Show that the system is consistent with infinitely many solutions by calculating the ranks using the method of minors.

2. If we add a fourth equation $x + y + az = b$, discuss what the relationship between $a$ and $b$ must be for the system to still have solutions.

A youth sports delegation in the Community of Madrid purchases public transport tickets for a championship. The tickets are: single ticket ($x$), 10-trip travel card ($y$), and monthly pass ($z$).

• The first group buys 2 single tickets, 1 10-trip travel card, and 1 monthly pass, for a total cost of $34$ €.

• The second group purchases 1 single ticket, 2 10-trip travel cards, and 1 monthly pass, paying a total of $42$ €.

• The third group buys 5 single tickets, 4 10-trip travel cards, and 3 monthly passes, and the invoice shows a total charged amount of $118$ €.

1. a) Model the situation using a system of linear equations $A \cdot X = B$.
2. b) Apply the Rouché-Frobenius Theorem to determine whether the system is consistent or if there is an error in the amounts invoiced by the transport company.
3. c) If there was an error in the third group's invoice, determine in a reasoned manner what exact amount they should have been charged for the system to be indeterminate consistent. Interpret what practical meaning it would have for the delegation if the system turned out to be indeterminate.

The following system of equations depending on the real parameter $k$ is given:

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

To discuss the consistency of the system according to the values of $k$, two students propose two different strategies:

• Carmen proposes using determinants: calculate $|A|$, find the values of $k$ that make it zero, and then study the ranks by bordering minors in the augmented matrix $A^*$.

• Daniel proposes applying Gaussian elementary row operations directly to the augmented matrix $(A|B)$ until it is in row echelon form in terms of $k$.

1. a) Critically compare both methods for this particular system. What algebraic or operational risks does the Gaussian method present when pivoting with expressions containing the parameter $k$ compared to the safety of calculating determinants?
2. b) Choose the method you consider most convenient and discuss the consistency of the system in detail for all real values of $k$ by applying the Rouché-Frobenius Theorem.

Consider the following system of linear equations depending on two real parameters $a$ and $b$:

$$\begin{cases} ax + y + z = 1 \\ x + ay + z = b \\ x + y + az = b^2 \end{cases}$$

Discuss the system using the Rouché-Frobenius Theorem according to all possible values of $a$ and $b$. In cases where the system is consistent with infinitely many solutions, state the degree of freedom (number of parameters in the solution) and describe the relative position of the three planes representing the equations.

Given the system of four equations in three unknowns:

$$\begin{cases} x + y + z = 2 \\ 2x + 3y + z = 3 \\ mx + 10y + 4z = 8 \\ 3x + 4y + 2z = k \end{cases}$$

1. a) Determine the values of the parameters $m$ and $k$ for the system to be consistent according to the Rouché-Frobenius Theorem.
2. b) Is there any value of $m$ that makes the system consistent with infinitely many solutions regardless of $k$? Justify your answer by analysing the ranks of the coefficient matrix and the augmented matrix.

An environmental engineering study analyses the flow of pollutants across three filtering stations ($A$, $B$ and $C$). The particle concentrations $x$, $y$ and $z$ at each station must satisfy the following technical equilibrium:

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

Discuss the consistency of the system depending on the real parameter $k$. In particular, analyse whether there is any value of $k$ for which the rank of the coefficient matrix is 1 and explain what this would mean geometrically for the filtering stations.

Let $A$ be a square matrix of order 3 such that its determinant is $|A| = m^2 - 4$. We consider the system of linear equations $A \cdot X = B$, where $B$ is a non-zero column vector.

It is also known that, for $m = 2$ and for $m = -2$, the matrix $A$ has rank 2, and that for $m = 2$ the system is consistent.

1. a) Discuss the consistency of the system according to the values of $m$ other than $-2$ and classify it (determinate or indeterminate) for $m \neq \pm 2$ and for $m = 2$.

1. b) If for $m = -2$ the system is inconsistent, what can we state about the rank of the augmented matrix $(A|B)$ in that case?

1. c) Describe a geometric situation (relative positions of the three planes) that corresponds to the case $m = 2$.

Consider the homogeneous system of equations:

$$\begin{cases} x + ky + z = 0 \\ kx - y + z = 0 \\ (k+1)x + (k-1)y + 2z = 0 \end{cases}$$

1. a) Prove that the system is consistent for any value of $k$.
2. b) Use the determinant of the coefficient matrix to find the values of $k$ that allow the existence of non-trivial solutions $(0,0,0)$.
3. c) For the values found in part b), determine the rank of the matrix and the number of parameters required to express the general solution.

In a transport logistics system, the routes between four cities are defined by a system of equations where the variables represent tonnes of cargo. The system is given by:

$$\begin{cases} x + y + z = a \\ x + (a+1)y + az = 2a + 1 \\ ax + y + z = a^2 \end{cases}$$

1. a) Discuss the system for the different values of $a$ using the Rouché-Frobenius Theorem.
2. b) If the system is consistent with infinitely many solutions, is it possible for the cargo $x$ to always be double the cargo $y$? Justify your answer.

Determine for which values of the parameters $p$ and $q$ the following system has:

1. A unique solution.

2. Infinitely many solutions.

3. No solution.

$$\begin{cases} x + 2y - z = q \\ px + (p+1)y + z = p \\ px + py + pz = p \end{cases}$$

For the case where it is consistent with infinitely many solutions, interpret whether the three planes intersect along a line or whether there are two coincident planes.

Consider the system of equations:

$$\begin{cases} x + y + mz = m \\ mx + (m-1)y + z = m \\ x + my + z = m \end{cases}$$

1. a) Calculate the value of $m$ that makes the rank of the coefficient matrix equal to 2.
2. b) For the value of $m$ found, study whether the system is consistent or inconsistent.
3. c) If it is consistent, can we assert that there exists a solution where $z = 0$? Reason your answer based on the structure of the system.

Design a system of 3 linear equations in 3 unknowns ($x, y, z$) that depends on a real parameter $\lambda$ and simultaneously satisfies the following three conditions:

1. For $\lambda = 1$, the system is inconsistent ($rg(A) = 2$ and $rg(A^*) = 3$).

2. For $\lambda = -2$, the system is indeterminate consistent with 1 degree of freedom (its general solution depends on a single real parameter).

3. For any $\lambda \notin \{1, -2\}$, the system is determinate consistent.

You must:

1. a) Explicitly write the three equations that make up your system, with their coefficients and constant terms in terms of $\lambda$.
2. b) Formally justify, by applying the Rouché-Frobenius Theorem (through the calculation of determinants and ranks), that your system exactly satisfies the three required conditions.

On an assessment test, the following theoretical question is asked:

"Let $S$ be a determinate consistent system consisting of 3 linear equations in 3 unknowns. If a fourth non-zero linear equation is added, analyze in a reasoned manner whether the new augmented system $S'$ can be: (i) Determinate Consistent, (ii) Indeterminate Consistent, or (iii) Inconsistent."

Javier answers the following:

"Since the resulting system has 4 equations and 3 unknowns ($m > n$), it is impossible for it to be determinate because there are redundant equations. Therefore, it can only be indeterminate consistent (if the new equation is a linear combination of the previous ones) or inconsistent (if it contradicts the first three)."

1. a) Identify the conceptual errors made by Javier and refute them by arguing from the Rouché-Frobenius Theorem and the theoretical bounds of the rank for matrices of size $4 \times 3$ and $4 \times 4$.
2. b) Determine which of the three options ((i), (ii), and (iii)) are mathematically possible and which are impossible. For each possible option, construct a concrete numerical example; for the impossible option, rigorously prove its impossibility based on the properties of rank.