College Linear Algebra, Please Help!

[jennybenny02]

This is a critical reading example in my book I'm trying to solve... Please help if you have any clue as to what Im writing about.

(Question)
Give an example of a matrix, A, for which the system Ax= b can be solved for any b, yet not always uniquely.


The first part of this assignment we are working with left and right inverses of a matrix, Im guessing it has something to do with that, but have no clue what to do for this problem or what to write

 

[soroban]

Hello, jennybenny02!

I have an answer, but it's not completely satisfactory . . .

Give an example of a matrix, \(\displaystyle A\), for which the system \(\displaystyle Ax\:=\:b\) can be solved for any \(\displaystyle b\), yet not always uniquely.

Let's limit our example to 2-by-2 matrices.

Assume: \(\displaystyle \;A\:=\:\begin{pmatrix}a & b\\ c & d\end{pmatrix},\;x\:=\:\begin{pmatrix}x \\ y\end{pmatrix},\;b\:=\:\begin{pmatrix}p \\ q\end{pmatrix}\)

Then we have the system: \(\displaystyle \;\begin{pmatrix}a & b \\ c & d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}\;=\:\begin{pmatrix}p\\q\end{pmatrix}\;\;\Rightarrow\;\;\begin{array}{cc}ax\,+\,by\:=\ \\ cx\,+\,dy\:=\:q\end{array}\)


The system will be <u>dependent</u> if: \(\displaystyle c\,=\,\frac{aq}{p}\) and \(\displaystyle d\,=\,\frac{bq}{p}\)

. . Then the second equation is: \(\displaystyle \;\frac{aq}{p}x\,+\,\frac{bq}{p}y\:=\:q\;\;\Rightarrow\;\;ax\,+\,by\:=\;p\)


Of course, \(\displaystyle p\,\neq\,0\) . . . but the problem said "for any \(\displaystyle b\)", didn't it?

 

[jennybenny02]

Thanks for the Help! I think I just didnt look at the problem hard enough, my brain is just mush right now. Probably because finals are in a week and I have 4 college math classes :shock:

Yes the problem asked for "any b" I think it might be asking for the case where its not always unique.

 

[jennybenny02]

Ok, another question... What about for non-square matrices?? EVen though the right or left inverse of a matrix that is not square is never unique. :shock: