Solve Ax=b Matrix System of Equations: Help Needed

  • Thread starter skujesco2014
  • Start date
  • Tags
    Matrix
In summary, the conversation discusses solving a system of equations of the type Ax=b, where A is a matrix and b is a vector. The conversation mentions a theorem that states that when the determinant of A is zero, the system has infinite solutions. The speaker also provides an example of parameterized solutions for the system. However, there is a mistake in the reduced row-echelon form of the system, which leads to a puzzle and discussion about elementary row operations.
  • #1
skujesco2014
24
0
Hi, all. I'm in desperate need of assistance with a matrix I can't get my head around of. I want to solve a system of equations of the type [itex]Ax=b[/itex], where

[tex] A=
\begin{pmatrix}
2 & 5 & -3 \\
1 & -2 & 1 \\
7 & 4 & -3
\end{pmatrix}
[/tex]

and where

[tex]b=
\begin{pmatrix}
-2 \\
-1 \\
-7
\end{pmatrix}
[/tex]

that is, [itex]b[/itex] is the negative of the first column. Written as it is above, [itex]A[/itex] has zero determinant and the determinant formed when the [itex]k^{th}[/itex] column of [itex]A[/itex] is substituted by the vector [itex]b[/itex] is clearly zero as well. A theorem says that in this case the system has infinite solutions. If one reduces the system to reduced row-echelon form the solutions can be parameterized as, for example, [itex]x_3=t, x_2=5t/9, x_1=t-1[/itex]. An immediate solution by inspection is [itex]x=(-1,0,0)^T[/itex] which one obtains letting [itex]t=0[/itex].

But let's give another value of [itex]t[/itex], for example, [itex]t=1[/itex] which gives [itex]x=(0,5/9,1)^T[/itex]. This is one of the parameterized solutions and yet it does not satisfy the original system. It does, however, satisfy the reduced system obtained from the original by gaussian elimination and should be equivalent, i.e.,

[tex]
\begin{cases}
x_1 - x_3 & = -1\\
9x_2 - 5x_3 & = 0
\end{cases}
[/tex]

But shouldn't my parameterized solution satisfy both original and reduced systems, no matter what? Yet, the only satisfying solution for the original system seems to be [itex]x=(-1,0,0)^T[/itex]. What am I not seeing here?

Thanks in advance.
 
Physics news on Phys.org
  • #2
I've discovered my mistake! I've sinned against the elementary row operations on matrix [itex]A[/itex]! The reduced system actually is

[tex]
\begin{cases}
x_1 -x_3/9& = -1\\
x_2-5x_3/9 & = 0
\end{cases}
[/tex]

The displayed erroneous reduced system above was obtained after mistakenly adding a multiple of a row to a multiple of another row. This isn't an allowed elementary operation. It was very intriguing that this little mistake lead to the puzzle that inspired this post.
 

1. How do you solve a matrix system of equations?

There are several methods for solving a matrix system of equations, including row reduction, Gaussian elimination, and Cramer's rule. However, the most common method is Gaussian elimination, which involves using a series of row operations to transform the matrix into row-echelon form and then back-substituting to find the values of the variables.

2. Can you explain the concept of row-echelon form?

Row-echelon form is a specific configuration of a matrix where all the leading coefficients (the first non-zero number) of each row are to the right of the leading coefficients of the rows above it, and any rows of all zeros are at the bottom. This form makes it easier to solve a matrix system of equations using back-substitution.

3. What is back-substitution and why is it used?

Back-substitution is the process of solving for the variables in a matrix system of equations by starting from the bottom row and substituting the values back into the equations above it. This is used because it allows for a simpler and more efficient way to solve for the variables compared to other methods such as solving for each variable individually.

4. Are there any special cases or exceptions when solving a matrix system of equations?

Yes, there are a few special cases that may arise when solving a matrix system of equations. These include inconsistent systems (where there is no solution), dependent systems (where there are infinitely many solutions), and matrices with zero rows or columns. In these cases, the solutions may need to be interpreted differently or may not exist at all.

5. Can you provide an example of solving a matrix system of equations?

Sure, let's say we have the following system of equations:
2x + 3y = 8
4x - 2y = 10
We can represent this system in matrix form as:
| 2 3 |
| 4 -2 |
To solve this system using Gaussian elimination, we would first use row operations to transform the matrix into row-echelon form:
| 1 3/2 |
| 0 -8 |
Then, using back-substitution, we can solve for the variables:
y = 1
x = 2
Therefore, the solution to the system is x = 2, y = 1.

Similar threads

  • Linear and Abstract Algebra
Replies
2
Views
248
  • Linear and Abstract Algebra
Replies
6
Views
359
  • Linear and Abstract Algebra
Replies
15
Views
1K
  • Linear and Abstract Algebra
Replies
11
Views
897
  • Linear and Abstract Algebra
Replies
14
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
731
  • Linear and Abstract Algebra
Replies
3
Views
866
  • Linear and Abstract Algebra
Replies
7
Views
791
  • Linear and Abstract Algebra
Replies
2
Views
919
  • Linear and Abstract Algebra
Replies
14
Views
2K
Back
Top