22.7. Consistency of a Linear System: Rank-Based Approach

Let us consider a linear system of equations:

AX=B

Where:

• A is the coefficient matrix,
• X is the column vector of unknowns, and
• B is the column vector of constants.

The equation AX = B represents a system of linear equations, where:

• 'A' represents the coefficients associated with the variables in your equations. For example, in the equation x + y = 3, 'A' would be [1, 1].
• 'X' is the vector that contains the variables (unknowns) that we are trying to solve for, such as [x, y].
• 'B' represents the constants on the right-hand side of the equations, which are the results of the equations. In our example, 'B' would be [3]. This setup allows us to manipulate the equations using matrix operations.

Imagine you're trying to distribute some resources, like drinks among friends during a party. If A represents how much each friend wants (like their drink preferences), X represents the number of drinks you need to allocate to each friend, and B represents the total number of drinks you have. You need to figure out how to match these preferences with the available drinks, just like we solve for 'X' in the equation.

The augmented matrix is

[A∨B].

The augmented matrix combines the coefficient matrix 'A' and the constant vector 'B' into a single matrix representation. This matrix allows for easier manipulation and helps us to apply row operations to analyze the system as a whole. It visually represents both the coefficients of the unknowns and the constants that the equations equal to.

Think of the augmented matrix as a complete list of ingredients and requirements for a recipe. For example, if you're making a cake, the matrix shows both the ingredients (coefficients) needed for each layer of the cake, while the constants tell you how many cakes you want to make.

The consistency and solution nature of this system depends on comparing the ranks of A and [A∨B].

To determine if the system of equations is consistent (meaning it has at least one solution), we compare the rank of matrix 'A' (the coefficient matrix) with the rank of the augmented matrix [A∨B]. The rank of a matrix is essentially the maximum number of linearly independent rows or columns within it. If both ranks are equal, the linear system is consistent. If they are not equal, the system is inconsistent, meaning there are no solutions.

Imagine you are a teacher trying to assign projects to students. If the number of projects (the rank of 'A') matches the number of available project outlines (the rank of [A∨B]), then all students can get a project. However, if you have more students than available outlines or if some outlines don’t match any project, you won’t be able to assign projects, similar to how a system becomes inconsistent.

Theorem: Rouché–Capelli Theorem

A system of linear equations is consistent (i.e., has at least one solution) if and only if

rank(A)=rank([A∨B])

If the system is consistent:

• Unique solution if rank(A)=number of variables
• Infinitely many solutions if rank(A)<number of variables

If the system is inconsistent: rank(A)≠rank([A∨B])

The Rouché–Capelli theorem provides a criterion for determining the consistency of a system of linear equations based on ranks. It specifies that a system is consistent only if the rank of the coefficient matrix 'A' is equal to the rank of the augmented matrix. If they are equal, the number of solutions can vary: if the rank equals the number of variables, there is a unique solution; however, if the rank is less than the number of variables, there are infinitely many solutions. Conversely, if the ranks are not equal, the system has no solutions.

Consider a popular restaurant's reservation system. If the number of tables (A) matches the number of reservations made (B), everyone gets seated smoothly. If there are more tables than reservations (rank(A) < rank([A∨B])), some tables will remain empty, indicating available opportunities. If reservations exceed tables (rank(A) ≠ rank([A∨B])), some customers will have to be turned away, representing an inconsistent situation.

Example 1: Consistent System Solve: $x + y + z = 6 \\ x + 2y + 3z = 14 \\ 2x + 3y + 4z = 20$

Coefficient matrix A:

[1 1 1] [ 6 ] 1 2 3 B= 14 2 3 4 20

Form augmented matrix and apply row operations:

[1 1 1 6 ] [A∨B]= 1 2 3 14 ⇒Reduce to REF 2 3 4 20

After reduction:

[1 1 1 6] 0 1 2 8 ⇒rank(A)=rank([A∨B])=2<3 0 0 0 0 ⇒ Infinitely many solutions

In this example, we have a system of equations that we solve step by step. The augmented matrix is formed by combining the coefficients and constants into a single matrix. After performing row operations to reduce it to Row Echelon Form (REF), we observe that there are two non-zero rows, indicating that the rank of matrix A is 2. Since the rank of [A∨B] is also 2 and is less than the number of variables (3), it shows there are infinitely many solutions. These solutions make sense as there’s not a single answer that satisfies all equations, but rather many combinations that work.

Imagine three friends trying to match their schedules for a weekend trip. If they have overlapping times for different plans (such as camping, hiking, or home), there is not one set way to heroically solve who can do what, just many options and alternatives available. This scenario mimics our equations that yield infinitely many solutions: we can have many combinations of activities, as long as their preferred times overlap.

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