Prove that the matrices have the same rank.

[Dafe]

Homework Statement

Prove that the three matrices have the same rank.


<br /> <br /> \left[<br /> \begin{array}{c}<br /> A\\<br /> \end{array}<br /> \right]<br /> <br />

<br /> <br /> \left[<br /> \begin{array}{c}<br /> A &amp; A\\<br /> \end{array}<br /> \right]<br /> <br />

<br /> <br /> \left[<br /> \begin{array}{cc}<br /> A &amp; A\\<br /> A &amp; A\\<br /> \end{array}<br /> \right]<br /> <br />

Homework Equations

The Attempt at a Solution

If elimination is done on the second matrix it will become:
<br /> <br /> \left[<br /> \begin{array}{c}<br /> A &amp; 0\\<br /> \end{array}<br /> \right]<br /> <br />

This means that the rank is still the same as A.

Elimination on the third matrix gives:
<br /> <br /> \left[<br /> \begin{array}{cc}<br /> A &amp; A\\<br /> 0 &amp; 0\\<br /> \end{array}<br /> \right]<br /> <br />

Since no new independent vectors are added, it also has rank A.

I do understand that this is so, but could someone please help me explain this mathematically?

Thanks.

 

[Billy Bob]

About the only thing different I would say is to suppose that when
<br /> \left[<br /> \begin{array}{c}<br /> A\\<br /> \end{array}<br /> \right]<br />
is reduced you obtain
<br /> \left[<br /> \begin{array}{c}<br /> R\\<br /> \end{array}<br /> \right]<br />


Then express your reduced forms of the other two matrices in terms of R instead of A. Refer to the number of nonzero rows in R, and you are done.

 

[Dafe]

Hi Billy Bob, thanks for the reply.
Here's the way I think you would do it: (just showing one matrix)

<br /> <br /> \left[<br /> \begin{array}{c}<br /> A &amp; A\\<br /> \end{array}<br /> \right]<br /> <br />

<br /> <br /> \left[<br /> \begin{array}{c}<br /> R &amp; 0\\<br /> \end{array}<br /> \right]<br /> <br />

# non zero rows = r for all matrices.