Rules for Solving Unknown Matrix A in Ax=b Problem?

[Genxi]

Homework Statement

I am trying to solve for the A matrix (3 x 3). I know matrix x is (3 x 1) and matrix b is (3 x 1), how do I go about solving for matrix A?

The Attempt at a Solution

I have not attempted this as I don't know the rules to initiate this problem.

Please offer me some hints or rules I should know on how to solve this problem


Thanks,


Genxi

 

[K29]

Well you could re-arrange it with matrix algebra:

$Ax=b$

$Axx^{T}=bx^{T}$

$A=\frac{bx^{T}}{xx^{T}}$

Note the last step is allowed because $xx^{T}$ is a scalar.(You can't divide matrices by matrices)

Familiarise yourself with the transpose of a matrix (in this case, a column vector) and matrix multiplication, and perhaps rules of matrix algebra and that should be all you need to understand the above.

This link may be helpful http://people.hofstra.edu/stefan_waner/RealWorld/Summary3.html , though it may go into much more depth than you need.



.

 

[DryRun]

Other than what K29 has suggested, here are some methods commonly used to solve systems of linear equations (matrix equations). You can do a little research, or refer in your book/s:

1. Cramer's rule.
2. Gauss elimination method (reduce to row echelon form)
3. LU-factorisation method
4. Using matrix inversion.

Personally, i prefer the Gauss elimination method which is quicker.

 

[Ray Vickson]

Well you could re-arrange it with matrix algebra:

$Ax=b$

$Axx^{T}=bx^{T}$

$A=\frac{bx^{T}}{xx^{T}}$

Note the last step is allowed because $xx^{T}$ is a scalar.(You can't divide matrices by matrices)

Familiarise yourself with the transpose of a matrix (in this case, a column vector) and matrix multiplication, and perhaps rules of matrix algebra and that should be all you need to understand the above.

This link may be helpful http://people.hofstra.edu/stefan_waner/RealWorld/Summary3.html , though it may go into much more depth than you need.



.

This is incorrect. Since x and b are column vectors, the objects xx^T and bx^T are 3×3 matrices, not scalars.

Anyway, if x and b are known but A is unknown, the equations Ax = b give 3 equations in the 9 unknowns a_ij, so the system is underdetermined. Additional information or some type of optimization criterion would need to be incorporated in order to obtain a unique solution.

RGV

 

[D H]

Well you could re-arrange it with matrix algebra:

$Ax=b$

$Axx^{T}=bx^{T}$

$A=\frac{bx^{T}}{xx^{T}}$

Note the last step is allowed because $xx^{T}$ is a scalar.(You can't divide matrices by matrices)

The last step is disallowed because $xx^{T}$ is a 3x3 matrix. What you might be able to do is post-multiply by the inverse of $xx^{T}$, but that too is disallowed because $xx^{T}$ is singular.

So let's go back to the start.

I am trying to solve for the A matrix (3 x 3). I know matrix x is (3 x 1) and matrix b is (3 x 1), how do I go about solving for matrix A?

You can't. Ax=b comprises three equations. However, you have nine unknowns, the nine elements of A. That's an underdetermined system. There are either no solutions or there are an infinite number of solutions.Edit
I now see that Ray Vickson beat me to it.

 

[K29]

My mistake. I was working too quickly. Apologies to OP