Prove that the matrices have the same rank.

  • Thread starter Dafe
  • Start date
  • #1
145
0

Homework Statement


Prove that the three matrices have the same rank.


[tex]

\left[
\begin{array}{c}
A\\
\end{array}
\right]

[/tex]

[tex]

\left[
\begin{array}{c}
A & A\\
\end{array}
\right]

[/tex]

[tex]

\left[
\begin{array}{cc}
A & A\\
A & A\\
\end{array}
\right]

[/tex]

Homework Equations





The Attempt at a Solution



If elimination is done on the second matrix it will become:
[tex]

\left[
\begin{array}{c}
A & 0\\
\end{array}
\right]

[/tex]

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

Elimination on the third matrix gives:
[tex]

\left[
\begin{array}{cc}
A & A\\
0 & 0\\
\end{array}
\right]

[/tex]

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.
 

Answers and Replies

  • #2
392
0
About the only thing different I would say is to suppose that when
[tex]
\left[
\begin{array}{c}
A\\
\end{array}
\right]
[/tex]
is reduced you obtain
[tex]
\left[
\begin{array}{c}
R\\
\end{array}
\right]
[/tex]


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.
 
  • #3
145
0
Hi Billy Bob, thanks for the reply.
Here's the way I think you would do it: (just showing one matrix)

[tex]

\left[
\begin{array}{c}
A & A\\
\end{array}
\right]

[/tex]

[tex]

\left[
\begin{array}{c}
R & 0\\
\end{array}
\right]

[/tex]

# non zero rows = r for all matrices.
 

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