Exercises: Gaussian elimination

Solve the following system of two equations in two unknowns using Gaussian elimination. Although you could use other methods from previous years, the objective is for you to practise transforming the coefficient matrix to obtain an echelon form:

$$\begin{cases} x + 2y = 8 \\ 3x - y = 3 \end{cases}$$

To help you, we illustrate the process of "making a zero" below the first diagonal entry:

In Gaussian elimination, the final step is back substitution. Solve the following system, which is already in echelon (triangular) form:

$$\begin{cases} x + y + z = 6 \\ 2y + 3z = 11 \\ 4z = 12 \end{cases}$$

Obtain the solution to the following system of linear equations using Gaussian elimination, performing the necessary elementary row operations to make the matrix upper triangular:

$$\begin{cases} x + y + z = 2 \\ 2x + 3y + 5z = 11 \\ x - 5y + 6z = 29 \end{cases}$$

Classify the following system according to its number of solutions after applying Gaussian elimination. State whether it is consistent with a unique solution, consistent with infinitely many solutions, or inconsistent, based on the last row obtained:

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

A system with infinitely many solutions (consistent undetermined) is one that has infinitely many solutions. Use Gaussian elimination to verify that the following system is of this type and write its general solution in terms of a parameter $\lambda$ (setting $z = \lambda$):

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

A graphic design studio equips its workstations with three orders from the same component supplier:

• First order: 1 keyboard, 1 mouse, and 1 mouse mat for $55\ \text{€}$.

• Second order: 2 keyboards, 2 mice, and 1 mouse mat for $102\ \text{€}$.

• Third order: 1 keyboard and 2 mouse mats for $51\ \text{€}$.

Set up the system of equations and calculate the unit price of each item using Gaussian elimination.

Given the following augmented matrix $(A|B)$ associated with a system of equations, perform the necessary elementary operations to obtain a row echelon matrix and solve the system:

$$(A|B) = \left( \begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 0 & 1 & -1 & 1 \\ 1 & 0 & 2 & 5 \end{array} \right)$$

Sometimes it is convenient to swap the order of the equations to simplify calculations (looking for "1" as a pivot). Rearrange and solve the following system using Gaussian elimination:

$$\begin{cases} 3x - y + 2z = 7 \\ 2x + y + z = 6 \\ x + y + z = 4 \end{cases}$$

Álvaro is solving a system of three linear equations with three unknowns using Gaussian elimination. After applying elementary row operations to the augmented matrix $(A|B)$, he obtains the following echelon form:

$$\begin{pmatrix} 1 & 2 & -1 & | & 3 \\ 0 & -1 & 4 & | & 1 \\ 0 & 0 & 0 & | & 2 \end{pmatrix}$$

Álvaro states on his exam: "Since the coefficient matrix in the last row only has zeros, one unknown vanishes; therefore, the rank of the matrix is 2, the system is consistent dependent, and it has infinitely many solutions that depend on a parameter $\lambda$".

1. a) Identify and explain Álvaro's conceptual error precisely. What is the difference between the rank of the coefficient matrix $A$ and the rank of the augmented matrix $A^*$ in this echelon matrix?
2. b) Write the algebraic equation with unknowns $x, y, z$ represented by the third row of the echelon matrix, and reason why this equation proves the inconsistency of the system without having to rely on memorized rules.
3. c) Correct Álvaro's conclusion by classifying the system according to the Rouché-Capelli theorem and interpret what happens geometrically with the three planes of the initial system.

Consider the following overdetermined system of three equations with two unknowns:

$$\begin{cases} x + y = 4 \\ 2x - y = 5 \\ 3x + 2y = 11 \end{cases}$$

1. a) Estimate: Before performing any numerical calculations, explain what each equation represents geometrically in the affine plane $\mathbb{R}^2$. If you draw three lines at random on a sheet of paper, what is more likely: that all three meet exactly at a single point or that they do not? Based on this geometric intuition, make a reasoned estimate of whether you expect the system to be consistent or inconsistent.
2. b) Calculate: Write the augmented matrix $(A|B)$ and apply Gaussian elimination to reduce it to row echelon form. Determine the ranks $\operatorname{rg}(A)$ and $\operatorname{rg}(A^*)$ from the echelon matrix.
3. c) Check: Solve the system if it is consistent or justify its inconsistency from the row echelon form. Compare the result with your initial estimate and explain what particular geometric relationship the three lines in this exercise share.

Solve the following system of linear equations using Gaussian elimination. If the system is consistent with infinitely many solutions, write the general solution in terms of a parameter $\lambda$:

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

Before starting, reflect on the choice of the "pivot" to make the calculations easier.

A technology company manufactures three models of graphics cards: GT, Pro, and Ultra. It is known that:

• The total cost of producing one unit of each model is $950 \text{ €}$.

• The cost of manufacturing two units of the GT model is equal to the sum of the costs of one unit of Pro and one unit of Ultra.

• If three units of Ultra are manufactured, the cost exceeds by $100 \text{ €}$ the cost of manufacturing one unit of each of the other two models together (GT and Pro).

Set up a system of equations and use Gaussian elimination to determine the production cost of each model.

Given the system depending on the real parameter $k$:

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

1. a) Apply Gaussian elimination to obtain a row echelon matrix.
2. b) Determine for which values of $k$ the system is consistent with a unique solution.
3. c) Solve it for $k = 0$.

Geometrically interpret the relative position of the three planes defined by the following equations, using Gaussian elimination to analyse the system:

$$\pi_1: x + 2y - z = 3$$ $$\pi_2: 2x - y + z = 1$$ $$\pi_3: 5x + 5y - 2z = 10$$

Do they intersect at a point, along a line, or do they have no common points? Justify your answer after transforming the augmented matrix.

A high-performance sports center organizes a summer camp. They have hired a total of $45$ instructors among specialists in swimming, athletics, and cycling.

• The number of swimming instructors is twice that of athletics.

• If they hired $5$ more athletics instructors, their number would equal that of cycling.

Use Gaussian elimination to find how many instructors there are in each specialty.

Consider the following homogeneous system of equations:

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

1. a) Prove, by reducing the coefficient matrix to echelon form, that the system admits non-trivial solutions (other than $x=0, y=0, z=0$).
2. b) Obtain the parametric solution of the system.

After partially applying Gaussian elimination to a system of three equations in three unknowns $(x, y, z)$, the following augmented matrix has been reached:

$$\begin{pmatrix} 1 & -1 & 2 & | & 3 \\ 0 & 1 & -1 & | & 1 \\ 0 & 0 & m^2-4 & | & m-2 \end{pmatrix}$$

Discuss the type of system (consistent with a unique solution, consistent with infinitely many solutions, or inconsistent) depending on the values of the parameter $m$. Is there any value of $m$ for which the system is inconsistent?

Three friends, Alba, Bruno, and Carlos, are saving for an end-of-year trip that costs $1200 \text{ €}$ per person.

• Between the three of them, they have saved half of the money needed for the trip of all three.

• Three times what Alba has minus what Carlos has is equal to what Bruno has.

• If Alba gave $100 \text{ €}$ to Carlos, both would have the same amount.

Set up the system and solve it using Gaussian elimination to find out how much each one has saved and how much they still need to be able to go on the trip.

A municipal urban transit company in a Spanish capital city analyzes transit ticket validations during a workday. Three ticket types are considered: Single Ticket ($x$), 10-Journey Travel Card ($y$), and Youth Monthly Pass ($z$), all measured in thousands of journeys.

The technical control system provides the following data:

1. The combined volume of journeys recorded across all three ticket types was $100$ thousand journeys.

2. The number of journeys made with the Youth Monthly Pass exceeded the sum of the journeys made with the Single Ticket and the 10-Journey Travel Card by $20$ thousand.

3. Validations with the 10-Journey Travel Card were exactly three times those made with the Single Ticket.

On the other hand, the financial audit reviews the revenue: a Single Ticket costs €$1,50$, each 10-Journey Travel Card journey yields €$0,80$, and each Youth Pass journey accounts for an average revenue of €$0,50$. The auditor's report states that the total revenue was $68$ thousand euros and suspects a discrepancy in the data.

1. a) Set up a system of 4 linear equations with 3 unknowns ($x, y, z$) that gathers all the available information.
2. b) Write the augmented matrix of the system (of size $4 \times 4$) and apply Gaussian elimination to reduce it completely to row echelon form.
3. c) Based on the ranks obtained via Gaussian elimination for the coefficient matrix and the augmented matrix, determine whether the auditor's report is consistent or inconsistent with the recorded validations. If there is an inconsistency, calculate what the exact actual revenue should have been for the data to be consistent.

To solve the following system of linear equations:

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

two approaches based on row operations are proposed:

• Approach A (Classical Gauss): Rearrange the rows to choose an optimal pivot, triangularize the augmented matrix by creating zeros below the main diagonal, and determine the solution using back-substitution.

• Approach B (Gauss-Jordan): Rearrange the rows and continue row operations to also create zeros above the main diagonal, transforming the coefficient matrix into the identity matrix to read the solution directly.

1. a) Justify which row swap is convenient to perform before starting operations to avoid working with fractions, and compare both approaches: how many row-elimination operations does Approach B perform that Approach A does not? In what practical situation is it worthwhile to take on the additional cost of Gauss-Jordan compared to classical Gauss?
2. b) Apply Approach A: show the augmented matrix step by step at each stage of the elimination and calculate the final solution using back-substitution.
3. c) If the system's vector of constant terms were replaced by $(0, 0, 0)$, explain without solving it whether it would be necessary to carry out the entire Gaussian elimination process to classify it and find at least one solution.

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

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

1. a) Discuss the system using the Gaussian elimination method according to the different values of $k$.
2. b) In the case where the system is consistent dependent (underdetermined), determine the general solution and find a particular solution such that the sum of its components is equal to 1.
3. c) Is there any value of $k$ for which the system represents three planes intersecting in a line? Justify your answer.

An aerospace engineering laboratory is designing a new alloy using three metals: Titanium ($x$), Aluminium ($y$), and Vanadium ($z$). The mass proportions must satisfy the following system, where $\alpha$ and $\beta$ are parameters depending on the smelting temperature:

$\begin{cases} x + y + z = \alpha \\ \alpha x + y + z = 1 \\ x + \alpha y + z = \beta \end{cases}$

1. a) Discuss the consistency of the system depending on the values of $\alpha$ and $\beta$.
2. b) If $\alpha = -1$, what value must $\beta$ have for the system to be consistent? Solve it in this case using Gaussian elimination.
3. c) Interpret the results obtained if the system were inconsistent from the perspective of alloy manufacturing.

A fintech company uses an algorithm to balance three investment portfolios. The capital flow between them is defined by the system:

$\begin{cases} x + 2y + 3z = 2 \\ 2x + 5y + 4z = -1 \\ x + 3y + az = b \end{cases}$

1. a) Apply Gaussian elimination to obtain a row echelon matrix and discuss the system according to the values of $a$ and $b$.
2. b) Determine the values of $a$ and $b$ so that the portfolios do not have a common equilibrium point (inconsistent system).
3. c) If $a = 1$ and $b = -3$, the system is consistent with infinitely many solutions. Find the solution that minimises the value of $x^2$.

Study the following system of 4 equations in 3 unknowns using Gaussian elimination:

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

1. a) Determine for which value of $a$ the system is consistent.
2. b) For this value of $a$, solve the system and indicate whether the solution is unique.
3. c) What relative position do the four planes defined by the equations occupy when the system is consistent?

Consider the homogeneous system dependent on the parameter $m$:

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

1. a) Use Gaussian elimination to find the values of $m$ for which the system admits solutions other than the trivial one $(0,0,0)$.
2. b) For the integer value of $m$ found in the previous section, solve the system and obtain the parametric expression of the solution.
3. c) Give a particular solution other than the trivial one where the unknowns are integers.

In a smart power grid, the current intensities $I_1, I_2, I_3$ and $I_4$ at four nodes satisfy the following Kirchhoff's laws, scaled by a resistance factor $R$:

$\begin{cases} I_1 + I_2 - I_3 + I_4 = 0 \\ 2I_1 - I_2 + I_3 = 5 \\ I_1 + I_3 - 2I_4 = 1 \\ 3I_1 + I_4 = 6 \end{cases}$

1. a) Solve the system using the Gauss–Jordan elimination method (achieving the maximum possible number of zeros in the matrix).
2. b) Is the system consistent with a unique solution? Justify whether it is possible to remove one of the measurements without losing information about the currents.

We wish to construct a system of three equations in three unknowns that simultaneously satisfies:

1. It is consistent with infinitely many solutions.

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

3. Upon applying Gaussian elimination, the last row of the echelon matrix is zero.

1. a) Propose an original system satisfying these conditions.
2. b) Explain step by step the process of reducing your proposed system to echelon form to show that it is indeed consistent with infinitely many solutions.

Consider the system of equations:

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

1. a) Discuss the system for the different real values of $a$.

1. b) If $a = 1$, describe the solution geometrically.

1. c) If $a = -2$, show using Gaussian elimination that the system is inconsistent. What geometric configuration do the three planes form in this case? Justify your answer with the relative positions of the planes (for example, by studying their normal vectors) and use a sketch drawn by yourself to help.

Design a system of three linear equations with three unknowns ($x, y, z$) that simultaneously satisfies the following four conditions:

1. None of the coefficients of the three unknowns in the first equation can be equal to 0, and its constant term must be $2$.

2. When applying Gaussian elimination to the augmented matrix $(A|B)$, using the coefficient $a_{11}$ as the first pivot to eliminate the first column, the resulting second row must be exactly $(0, 1, -2 \mid 1)$.

3. The system must be consistent dependent with one degree of freedom (rank 2), so that it geometrically represents three planes intersecting in a line $r$.

4. The solution line $r$ must have $\vec{v} = (3, 2, 1)$ as its direction vector.

1. a) Reasonably deduce the coefficients of the first equation from the orthogonality between the plane's normal vector and the direction of $\vec{v}$, respecting the imposed numerical constraints.
2. b) Construct the other two original equations of the system in such a way that, when applying elementary Gaussian operations, the indicated result is obtained in the second row and the third row becomes entirely zeros.
3. c) Write the initial augmented matrix of the system you formulated, apply Gaussian elimination step by step to verify that the required rows are reached, and obtain the parametric equations of the solution line, verifying that its direction vector matches $\vec{v}$.

Consider the system of equations dependent on the real parameter $a$:

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

To discuss the system using Gaussian elimination, Marcos sets up the augmented matrix and performs the following elementary row operations:

$$\begin{pmatrix} 1 & a & 1 & | & 1 \\ a & 1 & 1 & | & a \\ 1 & 1 & a & | & a^2 \end{pmatrix} \xrightarrow{\substack{F_2 \leftarrow F_2 - a F_1 \\ F_3 \leftarrow F_3 - F_1}} \begin{pmatrix} 1 & a & 1 & | & 1 \\ 0 & 1 - a^2 & 1 - a & | & 0 \\ 0 & 1 - a & a - 1 & | & a^2 - 1 \end{pmatrix}$$

Next, Marcos notices that in the second and third rows the factor $(1 - a)$ can be factored out, and computes:

$$F_2 \leftarrow \frac{1}{1-a} F_2, \quad F_3 \leftarrow \frac{1}{1-a} F_3 \implies \begin{pmatrix} 1 & a & 1 & | & 1 \\ 0 & 1 + a & 1 & | & 0 \\ 0 & 1 & -1 & | & -(a + 1) \end{pmatrix}$$

After this, he swaps $F_2 \leftrightarrow F_3$, eliminates the unknown in the last row, and concludes: "The only singular value is $a = -2$, where the last pivot becomes zero; for $a = 1$ there is no issue because the row has been simplified and the system is consistent independent".

1. a) Identify and explain the two conceptual errors made by Marcos in the use of Gaussian elimination with parameters (one regarding the algebraic validity of the operation performed and the other regarding the conclusion reached for $a = 1$).
2. b) Explain why dividing a row by an expression that depends on a parameter can destroy the equivalence of systems and what effect this produces on the solutions.
3. c) Carry out the correct and complete discussion of the system using Gaussian elimination, row-reducing the matrix without dividing by factors containing the parameter (or distinguishing the necessary previous cases), determining $\operatorname{rg}(A)$ and $\operatorname{rg}(A^*)$ for each $a \in \mathbb{R}$.
4. d) For the case $a = 1$, find the solution from the correct row-echelon matrix and geometrically interpret the relative positions of the three original planes.