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## 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.