Exercises: Homogeneous systems

A group of Year 13 students is analysing different types of mathematical models. They encounter the following system of equations:

$\begin{cases} 3x - 2y + z = 0 \\ x + 4y - 5z = 0 \\ 2x - y + 3z = 0 \end{cases}$

Give reasons to explain why this system is classified as a homogeneous system and check whether the triple $(x, y, z) = (0, 0, 0)$ is a solution to it. What is the technical name for this solution?

Geometrically, each equation of a system of three equations in three unknowns represents a plane in space. In the case of homogeneous systems, all the planes share a common characteristic regarding their position relative to the coordinate axes.

Observe the following graph representing three intersecting planes:

If the highlighted point is the origin of coordinates, is it possible for a homogeneous system to be inconsistent (have no solution)? Justify your answer based on the geometric interpretation.

Determine whether the following homogeneous system of two equations in two unknowns has only the trivial solution or if, on the contrary, it has infinitely many solutions:

$\begin{cases} 2x + 5y = 0 \\ 4x + 10y = 0 \end{cases}$

To do so, use the determinant of the coefficient matrix $A$.

Solve the following homogeneous system expressing its solutions in terms of two parameters (parametric form):

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

In a Mathematics Applied to the Social Sciences II exam, Hugo is faced with the following homogeneous system:

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

After calculating the determinant of the coefficient matrix and obtaining $|A| = 0$, Hugo writes the following conclusion:

“Since the determinant is 0, the rank of the coefficient matrix is less than 3. By the Rouché-Frobenius theorem, because the rank does not match the number of unknowns, the system has no solution; that is, it is an inconsistent system.”

1. a) Identify and explain in a reasoned manner the conceptual error made by Hugo. Mention which types of consistency a homogeneous system can exhibit according to the Rouché-Frobenius theorem and which solution is always guaranteed to exist.
2. b) Correct the resolution: calculate the rank of the coefficient matrix, classify the system appropriately, and find the general solution in parametric form.

Consider the following homogeneous linear system of two equations in three unknowns:

$$\begin{cases} 2x - 3y + z = 0 \\ 4x + y - 5z = 0 \end{cases}$$

1. a) Estimate and reason: Before carrying out any formal calculation, reason whether it is possible for this system to be consistent independent (that is, to admit only the trivial solution $(0, 0, 0)$). Base your estimation on the number of equations versus the number of unknowns and on the geometric interpretation of the equations as planes passing through the origin of coordinates.
2. b) Check: Apply the Rouché-Frobenius theorem by determining the ranks of the coefficient matrix and the augmented matrix. Solve the system by expressing the solutions in terms of a parameter $\lambda$ and confirm whether the obtained result matches the estimation you formulated in the previous section.

An architectural firm designs a structure based on three support beams whose internal stresses, represented by the variables $x$, $y$ and $z$, must be in equilibrium. The resulting mathematical model is the following homogeneous system:

$\begin{cases} x + y + z = 0 \\ 2x - y + 3z = 0 \\ 4x + y + 5z = 0 \end{cases}$

1. a) Justify, without solving the system, why it always admits at least one solution.
2. b) Calculate the rank of the coefficient matrix and determine whether the structure only has the "resting solution" (trivial) or if it allows other equilibrium configurations.
3. c) If non-trivial solutions exist, find their general expression.

In a chemistry laboratory, three components are mixed to stabilize a reagent. The relative proportions $x, y, z$ must satisfy the system:

$\begin{cases} x + y - z = 0 \\ x + 3y + z = 0 \\ 3x + 10y + 4z = 0 \end{cases}$

Show that the system is underdetermined (consistent with infinitely many solutions) and find its solutions in terms of a parameter $\lambda$. If the laboratory requires that $z = 5$, what would be the value of the other two components?

Study the rank of the matrix of the following homogeneous system and solve it in the cases where it is not consistent independent:

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

Give reasons to explain why this system has "more freedom" (more parameters) than a conventional consistent dependent system of three equations with a single redundant row.

Construct a homogeneous system of three linear equations in three unknowns that simultaneously satisfies the following conditions:

1. The rank of the coefficient matrix is 2.

2. One of the solutions is $(1, -1, 0)$.

3. The system is consistent dependent.

Subsequently, check that the designed system indeed has the trivial solution as one of its possible solutions.

As a financial analyst at a consulting firm, you need to formulate a homogeneous mathematical model with three unknowns ($x, y, z$) that simulates the accounting readjustment of three expenditure items without generating an overall imbalance. The system must simultaneously satisfy the following three requirements:

1. The triple $(x, y, z) = (2, -1, 3)$ must be a particular solution of the system.

2. The solution set must represent a line in three-dimensional space (that is, the system must be consistent dependent with a single degree of freedom).

3. The system must contain three linear equations such that none of them is zero ($0 = 0$) or identical to another.

1. a) Construct the three equations of the homogeneous system in a reasoned manner, justifying step by step how you ensure that the given particular solution is satisfied and that the rank of the coefficient matrix is exactly 2.
2. b) Solve the system you have constructed by expressing the general solution in terms of a parameter $\lambda$ and check for which exact value of $\lambda$ the solution $(2, -1, 3)$ is obtained.

The following homogeneous system of linear equations depending on the real parameter $k$ is proposed:

$$\begin{cases} x + y + kz = 0 \\ kx + y + z = 0 \\ x + ky + z = 0 \end{cases}$$

To determine the values of $k$ that cause the system to admit solutions other than the trivial solution, two students propose different methodological approaches:

• Gauss's Method (Álvaro): Row-reduce the coefficient matrix through elementary row operations until analyzing the presence of zeros on the main diagonal.

• Determinants Method (Beatriz): Calculate the determinant $|A|$, factor the resulting polynomial, and set it equal to 0 to identify when the rank drops below 3.

1. a) Critically evaluate both methods. Explain the algebraic difficulties and risks involved in Álvaro's method in this particular case (especially, the need to divide by expressions involving $k$ or distinguish cases when introducing zeros) in contrast to Beatriz's method, and justify which of the two is more reliable and faster.
2. b) Apply the method you selected to find all values of $k$ for which the system admits non-trivial solutions. For the smallest of these values, solve the system and write the general solution in parametric form.

Three technology companies, Alpha, Beta and Gamma, exchange consultancy services monthly. The cost of the services is offset in such a way that the net cash flow at the end of the quarter is zero for the global system. The balance of outstanding payments is modelled using the following homogeneous system, where $x, y, z$ represent the monthly payments of each company:

$$ \begin{cases} 2x - ay + z = 0 \\ ax - 4y + (a-2)z = 0 \\ x + 2y - z = 0 \end{cases} $$

Determine the value of the parameter $a$ for which the companies can make non-zero payments without breaking the system's equilibrium. For this value, calculate the ratio of proportionality between the payments of the three companies.

Let $M$ be a square matrix of order 3 whose rows are $F_1, F_2$ and $F_3$. We know that the homogeneous system $MX = 0$ is consistent dependent. If we perform an elementary transformation on the system such that the new matrix $M'$ has rows $F_1' = F_1 + F_2$, $F_2' = F_1 - F_2$ and $F_3' = k \cdot F_3$:

1. a) Under what conditions on the parameter $k$ does the new system $M'X = 0$ maintain the same degree of indeterminacy as the original system?

1. b) If the original system had as its solution the parametric line $x = \lambda, y = 2\lambda, z = -\lambda$, what will be the solution of the new system if $k=0$? Explain the geometric interpretation of this change.

A traffic control system in a low-emission zone of a Spanish city analyses the flow of vehicles in three main streets. The number of cars entering and leaving a central junction must satisfy the condition of zero net balance (entries = exits). The system is described by:

$$ \begin{cases} ax + y + z = 0 \\ x + ay + z = 0 \\ x + y + az = 0 \\ 2x + 2y + 2z = 0 \end{cases} $$

Analyse the consistency of this system of 4 equations in 3 unknowns depending on the parameter $a$. Is it possible for the system to be consistent dependent despite having more equations than unknowns? Justify your answer using the rank of the augmented matrix.

Find the values of $m$ and $n$ so that the following homogeneous system has a solution space of dimension 2 (that is, the solution depends on two free parameters):

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

For the values found, describe geometrically what the equations of the system represent and what set of points satisfies all the conditions.

Three agricultural cooperatives from Almería and Murcia ($C_1$, $C_2$, and $C_3$) share a common pool of resources and machinery. At the end of the year, they establish a closed internal exchange model (Leontief model) in which no money enters or leaves the consortium: the total value of what each cooperative consumes from the others must be exactly equal to the total value of what it produces. The distribution of consumption among them is as follows:

• $C_1$ consumes $20\,\%$ of its own production, $60\,\%$ of $C_2$'s production, and $10\,\%$ of $C_3$'s production.

• $C_2$ consumes $50\,\%$ of $C_1$'s production, $10\,\%$ of its own, and $20\,\%$ of $C_3$'s production.

• $C_3$ consumes $30\,\%$ of $C_1$'s production, $30\,\%$ of $C_2$'s, and $70\,\%$ of its own.

Let $p_1, p_2, p_3$ be the monetary values of the annual production of $C_1$, $C_2$, and $C_3$, respectively.

1. a) Formulate the homogeneous system that defines the accounting equilibrium of the three cooperatives (value produced = value consumed). Explain the economic meaning of the trivial solution $(0, 0, 0)$ and why it lacks practical utility for the cooperatives' economic activity.
2. b) Prove, without calculating the explicit determinant of the coefficient matrix, why the structure of the columns (whose sums are all equal to 1) guarantees that the system is consistent dependent.
3. c) Solve the system to obtain the production ratio $p_1 : p_2 : p_3$ at equilibrium. If cooperative $C_1$ produced goods worth $150.000$ euros, determine how much $C_2$ and $C_3$ must produce so that the consortium closes in financial equilibrium.

A risk management model for portfolios indexed to the IBEX 35 describes market stability through the following homogeneous system depending on two real parameters $a$ and $b$:

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

1. a) Inverse problem: Deduce the exact algebraic relationship that the parameters $a$ and $b$ must necessarily satisfy for the system to admit non-zero equilibrium configurations (that is, for infinitely many solutions to exist).
2. b) Parameter determination: Knowing also that one of the equilibrium portfolios observed in the market distributes its weights according to the ratio $(x, y, z) = (1, 1, 3)$, calculate the specific numerical values of parameters $a$ and $b$.
3. c) Resolution and geometric interpretation: With the values of $a$ and $b$ obtained in the previous section, find the general solution in parametric form and describe the relative position in space of the three planes defined by the equations of the system, justifying whether they intersect at a single point, along a line, or if two of them are coincident.

Topic challenge. How does the electricity generation mix affect the bill? Your mission is to advise an energy cooperative based on real data from Red Eléctrica de España (REE).

1. Data collection and model: Check the daily generation (GWh) of three sources (solar, wind, and hydro) over three days on REE. Model the cost allocation setting the matrix expression $A \cdot X = B$, and introduce a parameter $k$ to simulate a variable ecological tax.

2. Discussion and resolution: Apply the Rouché-Frobenius theorem to classify the solutions of the system according to the values of $k$. For the base case ($k = 1,0$), calculate the solutions applying the Gaussian elimination method and corroborate the result using Cramer's rule. Also set up the associated homogeneous system $A \cdot X = 0$ and interpret its economic meaning.

3. Analysis and final product: Assuming that daily demand follows an $N(\mu, \sigma)$ distribution, calculate its statistical parameters with a spreadsheet or GeoGebra from your sample. Write a brief report with a reasoned proposal for the cooperative.

Assessment criteria:

• Rigor in the matrix expression and in the discussion with Rouché-Frobenius according to the parameter.

• Correctness in applying and comparing the Gaussian elimination method, Cramer's rule, and the homogeneous system.

• Precision when calculating the parameters of the normal distribution with real data using GeoGebra or a spreadsheet.