Rouché-Frobenius Theorem

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.

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 $rg(A^*) = 2$.

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).

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

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.