# General Solution to a Singular System (or no solution)

• MurdocJensen
In summary, if you have a square, noninvertible matrix (like A in the example), then reducing the matrix will always yield at least one row of zeroes. If you have an augmented matrix where the last column is not 0, then there exist an infinite number of solutions.
MurdocJensen
When I realize that I am going to have a singular matrix (after exhausting row swap options and maybe even some elimination steps) what about the matrix tells me whether or not I can have a general solution?

Your question is sparse on details, so I'll answer what I think you're asking. Let's say we have these matrix equations:
Ax = 0
Ax = b

In both equations I'm assuming that A is a square, noninvertible matrix (i.e., |A| = 0).

Because |A| = 0, row reducing A will yield at least one row whose entries are all zero. This means that we have a system of equations with fewer equation than variables, meaning that at least one variable is free, so there are an infinite number of solutions for x.

If we represent the second matrix equation by an augmented matrix, row reducing A will still yield at least one row of zeroes on the left side of the augmented matrix. If the element of b that corresponds to that zero row is not zero, that row of the augmented matrix represents the equation
0x1 + 0x2 + ... + 0xn = bk, which has no solution. In this case, the system of equations is inconsistent.

Nothing about the coefficient matrix tells you that! If you are talking about the "augmented" matrix, where you have added the vector "b" (assuming your equation is Ax= b) as an additional column to the coefficient matrix, then the fact that the coefficient matrix is singular tells you that a row reduction will reduce the last row of the coefficient matrix to all "0"s. If there exist a row of the row-reduced augmented matrix where all entries except the last are "0" and the last column is not, there is NO solution. If, whenever all entries in a row, up to the last column, are "0" the last column entry in that row is also, then there exist an infinite number of solutions (so you can find a "general solution").

quickreply: i want to post an example system. anyone know how to type and copy matrices?

The fact that a matrix is singular, meaning it has no inverse, indicates that the system of equations it represents has either no solution or an infinite number of solutions. In order to determine which of these scenarios is the case, we can look at the reduced row echelon form (RREF) of the matrix.

If the RREF of the matrix has a row of all zeros, then the system has no solution. This is because in the RREF, this row represents an equation such as 0x + 0y + 0z = k, where k is a non-zero number. This is impossible to satisfy, as any value for x, y, or z will result in 0 on the left side of the equation, not k.

On the other hand, if the RREF of the matrix has no rows of all zeros, then the system has an infinite number of solutions. This is because in the RREF, each non-zero row represents an equation such as x + 0y + 0z = k, where k is a constant. This means that x can take on any value, and y and z are free variables. Therefore, there are infinitely many solutions to the system.

In summary, the RREF of a singular matrix can provide information about the existence and nature of solutions to the corresponding system of equations. It is important to carefully analyze the RREF in order to determine if the system has no solution or an infinite number of solutions.

## 1. What is a singular system?

A singular system is a system of equations where there is no unique solution. This means that there are either infinitely many solutions or no solutions at all.

## 2. How can you tell if a system is singular?

A system is singular if the determinant of its coefficient matrix is equal to zero. This means that the system has either no solutions or infinitely many solutions.

## 3. Can a singular system have more than one solution?

Yes, a singular system can have infinitely many solutions. This occurs when the equations in the system are not independent and are essentially describing the same relationship.

## 4. What is the difference between a singular system and a system with no solution?

A singular system has either infinitely many solutions or no solutions, while a system with no solution has no solutions at all. In other words, a singular system has some sort of solution, while a system with no solution has no solution at all.

## 5. How can a singular system be solved?

A singular system cannot be solved using traditional methods, such as substitution or elimination. Instead, it can be solved by finding the general solution, which represents all possible solutions for the system. This can be done by setting one variable as a parameter and expressing the other variables in terms of that parameter.

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