Having difficulty with an Ax=b problem and understanding it

  • Thread starter blhhblah
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In summary, when dealing with matrices, it is important to construct the augmented matrix A|b and perform row reduction to obtain the reduced-row echelon form. From there, the solutions for the variables can be extracted directly from the matrix. However, if there are more unknowns than equations, the matrix may be inconsistent and there will be no solutions. In this case, the least squares method can be used to obtain an approximate solution.
  • #1
blhhblah
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I'm having trouble with Ax=b matrices. I get really confused when dealing with matrices that have a different number of rows from columns. An example is this problem:
(A is a 3x2 matrix)
A =
3 1
4 2
-5 -1

(b is a 3x1 matrix)
b =
1
0
-3
I need to find the solution set for x. Aren't there only 2 variables?
Thanks.
 
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  • #2
Yes there are only two variables but 3 equations. Construct the augmented matrix A|b and perform row reduction until you get the reduced-row echelon form of the matrix. Now you can easily extract the solutions.
 
  • #3
is this the reduced row echelon form?

1 0 1
0 1 -2
0 0 0
 
  • #4
regarding another question...

how do you solve for x given A is a (3x4) matrix and b is a (4x1) matrix?
 
  • #5
Yeah I got that as well. Now you can read the solutions for the variables directly from the matrix.
 
  • #6
blhhblah said:
regarding another question...

how do you solve for x given A is a (3x4) matrix and b is a (4x1) matrix?
It's by using the same technique as well. But note that this time you have more unknowns than equations. What does that tell you about the nature of the solutions you will have?
 
  • #7
i guess all the values won't be exact, but I'm not sure how to set this up
 
  • #8
What do you mean by "exact"? Just write out the augmented matrix and again use row-reduction as before. Note that it is entirely possible that the matrix might be inconsistent, in that case there are no solutions.
 
  • #9
matrix A was

2 8 4 16
1 1 2 2
1 -1 2 -2

I was able to reduce it to
1 0 2 0
0 1 0 2
0 0 0 0

but I'm not sure how to match it up to b, which is

-1
9
5
17
 
  • #10
Your first matrix was equaivalent to the set of equations
2x+ 8y+ 3z= 16, x+ y+ 2z= 2, and x- y+ 2z= -2 with b= <x, y, z>.

You have reduced it to a matrix equivalent to
x+ 2z= 0 and y= 2. What possible values of x, y, and z satisfy both of those?
 
  • #11
Whenever I have more or fewer equations than unknowns, I use the following to obtain the least squares answer. [itex]A^T A x = A^T b[/itex], where [itex]A^T[/itex] is the transpose of A.
 

What is an Ax=b problem?

An Ax=b problem is a system of linear equations, where A is a matrix of coefficients, x is a vector of unknown variables, and b is a vector of constants. The goal is to find a solution for x that satisfies all of the equations in the system.

Why am I having difficulty with an Ax=b problem?

There could be several reasons for difficulty with an Ax=b problem. It could be due to a lack of understanding of the concepts involved, a mistake in the calculations, or a complex system with no clear solution.

How can I better understand an Ax=b problem?

To better understand an Ax=b problem, it is important to have a strong foundation in linear algebra and matrix operations. It may also be helpful to break down the problem into smaller, more manageable parts and practice solving similar problems.

What are some common mistakes when solving an Ax=b problem?

Some common mistakes when solving an Ax=b problem include errors in calculations, not considering all of the equations in the system, and not using the correct method or algorithm for solving the problem.

What resources can I use to help with an Ax=b problem?

There are many resources available to help with an Ax=b problem, including textbooks, online tutorials, practice problems, and working with a tutor or classmate. It is also helpful to seek clarification from your instructor or teaching assistant if you are still having difficulty.

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