Rouché-Frobenius Theorem
At the corner store, Ximena keeps three receipts for sandwiches, juices, and chips. She wants to find the price of each item, but she suspects the third receipt doesn't tell her anything new. How can she know, without solving anything, whether three equations are enough to find three prices? With two equations you compared ratios between coefficients. With three or more you need a more general tool: the rank of a matrix. The Rouché-Frobenius theorem compares two ranks and tells you what type of system you have without solving it.
The rank of a matrix
A row is a combination of others when it adds no new information. That is exactly what happens with Ximena's redundant receipt.
You can also compute the rank with minors: if there is a nonzero minor of order 3, the rank is 3. If $\det A=0$ but some minor of order 2 is nonzero, the rank is 2. For example, in the matrix above $\det A=0$ and the minor $\begin{vmatrix}1&1\\2&3\end{vmatrix}=1\neq 0$, which confirms $\operatorname{rg}(A)=2$.
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…
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.
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…
Consider the system:
$$\begin{cases} x + y + z = 3 \\ 2x - y + z = 2 \\ x - 2y = -1 \end{cases}$$-
Show that the system is consistent with infinitely many solutions by calculating the ranks using the method of minors.
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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.
The Rouché-Frobenius theorem
For a system, there are two matrices: the coefficient matrix $A$ and the augmented matrix $A^*=(A\mid B)$, which adds the column of constant terms.
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…
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 $n = 3$:
- a) $rg(A) = 3$ and $rg(A^*) = 3$.
- b) $rg(A) = 2$ and $rg(A*) = 3$.
- c) $rg(A) = 2$ and $rg(A^*) = 2$.
Discuss the nature of the system formed by the equations of three lines in the plane, knowing that the system is given by: $$\begin{cases} 2x + 3y = 5 \\ x - y = 1 \\ 4x…
Discuss the nature of the system formed by the equations of three lines in the plane, knowing that the system is given by:
$$\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?
Classifying without solving
The theorem lets you draw conclusions using only the size of the system.
Fewer equations than unknowns. If there are $m Homogeneous systems. If all the constant terms are 0, $A^*$ only adds a column of zeros, so $\operatorname{rg}(A^*)=\operatorname{rg}(A)$. The system is always consistent, because $(0,0,\dots,0)$ is a solution (the trivial solution). Moreover: If $\operatorname{rg}(A)=n$, only the trivial solution exists. If $\operatorname{rg}(A) For a square homogeneous system, this is equivalent to: nontrivial solutions exist if and only if $\det A=0$. An "influencer" is analyzing 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 realized that there are more unknowns than equations: Where $x, y, z$ is the investment in pesos 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? 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: Study the consistency of the following homogeneous system as a function of the parameter $m$: State for which values of $m$ the system admits non-trivial solutions $(0, 0, 0)$.
An "influencer" is analyzing the growth of their followers across three different social networks (TikTok, Instagram, and Twitch). They have set up a system of equations…
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…
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…
Discussion with parameters
When the system depends on a parameter, to discuss it means to say what type of system results for each value, without solving it. The plan, for square systems:
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Compute $\det A$ as a function of the parameter.
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The values that do not make it zero give $\operatorname{rg}(A)=n$, and therefore a unique solution.
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For each value that makes it zero, substitute and compute $\operatorname{rg}(A)$ and $\operatorname{rg}(A^*)$ separately.
If the system is not square (more or fewer equations than unknowns), there is no $\det A$. You compute the ranks using minors, or by row reducing the augmented matrix and analyzing each value of the parameter that changes the number of nonzero rows. If the parameter appears only in the constant term, $A$ does not change and only $\operatorname{rg}(A^*)$ varies.
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…
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).
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…
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 following system of linear equations depending on two real parameters $a$ and $b$: $$\begin{cases} ax + y + z = 1 \\ x + ay + z = b \\ x + y + az = b^2…
Consider the following system of linear equations depending on two real parameters $a$ and $b$:
$$\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.
Geometric interpretation: three planes
Each equation $ax+by+cz=d$ is a plane in space. Solving the system means finding the points common to all three planes. The ranks tell you how they are arranged:
| $\operatorname{rg}(A)$ | $\operatorname{rg}(A^*)$ | Position of the planes |
|---|---|---|
| 3 | 3 | They intersect at one point |
| 2 | 2 | They intersect along a line (or two coincide and the third cuts them) |
| 2 | 3 | No common point: triangular prism, or two parallel planes cut by the third |
| 1 | 1 | All three planes are the same |
| 1 | 2 | No common point: parallel planes (or two coincident and one parallel) |
To decide between the two options in the same row, check whether the rows of $A^*$ are proportional in pairs. If two rows are proportional in $A$ but not in $A^*$, those planes are distinct parallel planes. If they are proportional in the full $A^*$, those planes coincide. If no pair is proportional and $\operatorname{rg}(A^*)=3$ with $\operatorname{rg}(A)=2$, it is a triangular prism (they intersect two at a time, but there is no common point).
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…
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$):
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The three planes intersect at a single point.
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Two planes are parallel and the third intersects them.
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The three planes intersect along a common line.
Match each case with one of these options:
- a) $rg(A) = 2, rg(A^*) = 2$.
- b) $rg(A) = 3, rg(A^*) = 3$.
- c) $rg(A) = 2, rg(A^*) = 3$.
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$$…
Consider the three planes represented in space whose equations are given below:
$$\pi_1: x + ay + z = 1$$ $$\pi_2: ax + y + (a-1)z = a$$ $$\pi_3: x + y + z = a + 1$$Study the relative position of the three planes as a function of the parameter $a$ based on the rank of the coefficient matrix $A$ and the augmented matrix $A^*$.
Determine for which values of the parameters $p$ and $q$ the following system has: 1. A unique solution. 2. Infinitely many solutions. 3. No solution. $$\begin{cases} x…
Determine for which values of the parameters $p$ and $q$ the following system has:
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A unique solution.
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Infinitely many solutions.
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No solution.
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.
Back to the beginning: if Ximena's third receipt is the sum of the other two, its row becomes zero when reduced, so $\operatorname{rg}(A)=2<3$. With three prices to find, that is not enough: there will not be a unique price for each item.