Finding System Solutions of the system Ax=0; A being a matrix

In summary, the task was to find all solutions of the system Ax=0, where A is a 3x3 matrix, and the approach involved turning x into a 3x1 matrix and solving the system of equations. The result was that each value of x was equal to 0, which is the only solution when A is invertible.
  • #1
gpax42
25
0

Homework Statement



Find all solutions of the system Ax=0
where A = the 3x3 matrix

[1 3 2]
[2 6 9]
[2 8 8]

Homework Equations



not really sure what equations to include

The Attempt at a Solution



wasnt positive how to go about answering this question because I am not sure what its asking for...

i turned "x" into a 3x1 matrix consisting of x1, x2 and x3 and tried to solve the system of equations...

1x1 + 3x2 + 2x3 = 0
2x1 + 6x2 + 9x3 = 0
2x1 + 8x2 + 9x3 = 0

after trying to solve this as a linear system of equations problem, i ended getting that each value (x1, x2, x3) was equal to 0... is this anywhere close to a correct approach to this problem? Thanks for any help you have to offer :smile:
 
Physics news on Phys.org
  • #2
Yes, since this matrix is indeed invertible.
 
  • #3
Do you know about row operations for reducing a matrix? If you do, using them to reduce your matrix would be simpler than solving three equations in three unknowns.
 
  • #4
My class learned about row operations today but I didn't think to apply them because I began this problem yesterday...

Thanks for your advice :biggrin:
 
  • #5
If A is invertible, then the only solution to Ax= 0 is x= 0.
 
  • #6
gpax42 said:
after trying to solve this as a linear system of equations problem, i ended getting that each value (x1, x2, x3) was equal to 0... is this anywhere close to a correct approach to this problem? Thanks for any help you have to offer :smile:

This is a linear system of equation which is equivalent to the matrix representation of the problem. And yes, your solution is correct, as far as I checked.
 

FAQ: Finding System Solutions of the system Ax=0; A being a matrix

How do I find the solution to a system of equations with a matrix?

The solution to a system of equations with a matrix can be found by setting the matrix equal to the zero vector and then solving for the variables using row reduction or Gaussian elimination.

Can a system of equations with a matrix have more than one solution?

Yes, a system of equations with a matrix can have infinitely many solutions or no solution at all. This depends on the properties of the matrix, such as its rank and determinant.

What is the importance of finding system solutions with a matrix?

Finding system solutions with a matrix is important in many fields of science, such as physics, engineering, and computer science. It allows us to solve complex systems of equations and model real-world problems.

Are there any special methods for finding system solutions with a matrix?

Yes, there are several special methods for finding system solutions with a matrix, such as Cramer's rule, inverse matrices, and the Jordan canonical form. These methods can be used to solve specific types of systems or matrices.

How can I check if my solution to a system of equations with a matrix is correct?

You can check your solution by plugging it back into the original system of equations. If it satisfies all the equations, then it is a correct solution. Additionally, you can use matrix algebra to verify if your solution is correct.

Similar threads

Matrix Relative to B and B' R3 to R3
Replies
1
Views
1K
How to find a system of equations when the solution is given?
Replies
6
Views
560
Is the Matrix Symmetric Positive Definite for Cholesky Decomposition?
Replies
2
Views
2K
System of equations: finding a plane
Replies
7
Views
2K
Solving a system of differential equations by fundamental matrix
Replies
8
Views
2K
Is there a better way to find the number of integer solutions
Replies
5
Views
2K
Does there exist a 2x2 non-singular matrix with only one 1d eigenspace?
Replies
2
Views
946
Find the basis of a vector subspace of R^2,2
Replies
9
Views
3K
Proving stability of linear system equations
Replies
2
Views
725
Show that the system of equations has no unique solution
Replies
2
Views
586

Hot Threads

Recent Insights

Back
Top