Solve Matrix Equations: Ax=b | Help with A, b, x

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In summary, the conversation discusses a system of equations in the form of Ax=b, where A and b are given matrices. It is noted that when b is the negative of the first column of A, the system has infinite solutions. By reducing the system to reduced row-echelon form, the solutions can be parameterized. However, it is observed that not all parameterized solutions satisfy both the original and reduced systems. The conversation concludes by discussing the kernel of the matrix and how scaling by a factor can result in different solutions.
  • #1
skujesco2014
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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.
 
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  • #2
I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 
  • #3
The vector

[tex]
\left(\begin{array}{c}
1 \\ 5 \\ 9 \\
\end{array}\right)
[/tex]

is contained in the kernel of the mentioned matrix, and all other elements of the kernel can be obtained by scaling this. It seems you have attempted to scale this by a factor [itex]\frac{t}{9}[/itex] with a real coefficient [itex]t[/itex], but you have made a mistake with the [itex]x_1[/itex] component.
 
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1. What is a matrix equation?

A matrix equation is an equation in which a matrix is equal to another matrix or a vector. It is used to solve systems of linear equations and is written in the form of Ax=b, where A is the coefficient matrix, x is the variable matrix, and b is the constant matrix.

2. How do you solve a matrix equation?

To solve a matrix equation, you can use the Gauss-Jordan elimination method or the inverse matrix method. In the Gauss-Jordan method, you reduce the matrix to row echelon form and then back-substitute to find the values of the variables. In the inverse matrix method, you multiply both sides of the equation by the inverse of the coefficient matrix.

3. What is the purpose of using matrices in equations?

Matrices are used in equations to represent systems of linear equations in a compact and organized way. They also allow for efficient calculation of solutions using various methods such as Gauss-Jordan elimination and matrix inversion.

4. What are the requirements for a matrix equation to have a solution?

A matrix equation has a solution if the coefficient matrix is square and its determinant is non-zero. This ensures that the system of equations is consistent and has a unique solution.

5. What is the role of the constant matrix in a matrix equation?

The constant matrix, also known as the right-hand side (RHS) matrix, contains the constants on the right side of the equation. It represents the values that the system of equations is equal to and is used in the calculation of the solution to the equation.

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