Register to reply

Help with a matrix

by skujesco2014
Tags: matrix
Share this thread:
skujesco2014
#1
Jul26-14, 03:24 PM
P: 16
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.
Phys.Org News Partner Mathematics news on Phys.org
Researcher figures out how sharks manage to act like math geniuses
Math journal puts Rauzy fractcal image on the cover
Heat distributions help researchers to understand curved space
Greg Bernhardt
#2
Aug3-14, 12:02 PM
Admin
Greg Bernhardt's Avatar
P: 9,711
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?
jostpuur
#3
Aug3-14, 01:01 PM
P: 2,068
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.


Register to reply

Related Discussions
Defining matrix of variables in mathematica and solving matrix differential equation Math & Science Software 4
Variance-covariance matrix and correlation matrix for estimated parameters Set Theory, Logic, Probability, Statistics 0
Real matrix - eigenvector matrix R - diagonal eigenvalue matrix L Programming & Computer Science 0
[Matrix Algebra] Special matrix, columns as a first order derivatives Calculus & Beyond Homework 1
Prove that Hermitian/Skew Herm/Unitary Matrix is a Normal Matrix Calculus & Beyond Homework 2