If A is singular; solution space of Ax=b

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In summary: No.2) Yes.3) Yes.3):The answer to this question is "Yes". First, take the matrix A and reduce it to row-echelon form. This will give you a matrix in which all the eigenvalues are zero. Next, use the Inverse Matrix Theorem to find all the solutions to the system Ax=0.
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Bipolarity
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Suppose it is known that A is singular. Then the system Ax=0 has infinitely many solutions by the Invertible Matrix theorem.

I am curious about the system Ax=b, for any column vector b. In general, i.e. for all vectors b, will this system be inconsistent, or will it have infinitely many solutions?

Surely there exists a vector b for which this system is inconsistent. For otherwise, if it were consistent for every vector b, it would necessarily be invertible (again by the IM theorem), but by assumption it is not.

So here are a few questions I have begun to think about, but not fully able to explain:

Given that A is a singular square matrix:

1) Does there necessarily exist a vector b for which Ax=b has infinitely many solutions?
2) Does there necessarily exist a vector b for which Ax=b has a unique solution?
3) If b is a certain column vector, can one determine simply by inspection whether Ax=b is inconsistent, has a unique solution, or has infinitely many solutions?

I would appreciation an answer to these questions. I prefer to do the "proofs" myself, so don't give anything away. Thanks much!

BiP
 
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  • #2
1) Yes.
2) No.
3) Yes.
 
  • #3
3):
"simply" - yes
"by inspection" - try it on a 100 x 100 matrix and see if you still think the answer is still "yes".
 
  • #4
Think of what happends when Gauss-Jordan elimination is applied to the augmented matrix of the system, and compare with the coefficient matrix. What happends with pivot positions, free variables, etc.?
 
  • #5
Hey all, I have been able to figure out why the answer to the second question is "No".
Also, I have been able to figure out how you can determine whether the solution set is null or whether it is infinite once the matrix A has been reduced to row-echelon form.

But is it possible to determine the solution space of Ax=b without applying Gaussian elimination?

BiP
 

1. What does it mean if A is singular?

If A is singular, it means that the matrix A does not have an inverse. This means that there is no unique solution to the equation Ax=b.

2. Can a singular matrix have a solution space?

Yes, a singular matrix can have a solution space. The solution space of a singular matrix is either empty or infinite. If the solution space is infinite, it means that there are multiple solutions that satisfy the equation Ax=b.

3. How is the solution space of a singular matrix different from a non-singular matrix?

The solution space of a singular matrix is different from a non-singular matrix in that a singular matrix does not have a unique solution. A non-singular matrix has a unique solution and its solution space is a single point.

4. Can a singular matrix be used to solve a system of equations?

Yes, a singular matrix can be used to solve a system of equations. However, the solution space may be infinite and there may be multiple solutions that satisfy the equations.

5. How can we determine if a matrix is singular or non-singular?

A matrix is singular if its determinant is equal to 0. To determine if a matrix is singular or non-singular, we can calculate the determinant using various methods such as cofactor expansion, row reduction, or using software. If the determinant is 0, the matrix is singular. If the determinant is not 0, the matrix is non-singular.

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