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.

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.

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.

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.

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).

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.

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.

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.