If A is singular; solution space of Ax=b


by Bipolarity
Tags: singular, solution, space
Bipolarity
Bipolarity is offline
#1
Dec14-12, 01:08 PM
P: 783
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
Phys.Org News Partner Science news on Phys.org
Better thermal-imaging lens from waste sulfur
Hackathon team's GoogolPlex gives Siri extra powers
Bright points in Sun's atmosphere mark patterns deep in its interior
Michael Redei
Michael Redei is offline
#2
Dec14-12, 02:03 PM
P: 181
1) Yes.
2) No.
3) Yes.
AlephZero
AlephZero is offline
#3
Dec14-12, 04:02 PM
Engineering
Sci Advisor
HW Helper
Thanks
P: 6,342
3):
"simply" - yes
"by inspection" - try it on a 100 x 100 matrix and see if you still think the answer is still "yes".

Erland
Erland is offline
#4
Dec14-12, 06:48 PM
P: 302

If A is singular; solution space of Ax=b


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.?
Bipolarity
Bipolarity is offline
#5
Dec15-12, 01:09 PM
P: 783
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


Register to reply

Related Discussions
Series Solution around singular point Calculus & Beyond Homework 1
Singular solution for a differential equation? Calculus & Beyond Homework 0
General Solution to a Singular System (or no solution) Linear & Abstract Algebra 3
singular solution Calculus & Beyond Homework 1
Singular Solution Differential Equations 3