# ERO's and ECO's

• December 30th 2010, 09:13 AM
worc3247
ERO's and ECO's
Let A be an $m \times n$ matrix with entries in $\mathbb{R}$. Show by using elementary row and column operations, that there are invertible matrices P and Q (where P has size $m \times m$ and Q has size $n \times n$) such that:
$PAQ=$\left( \begin{array}{cc} I_r & 0 \\ 0 & 0 \\ \end{array} \right)$$.

I can sort of see why this is true. You would need to use ECO's to make all but 1 entries in a row 0, and then permute the rows until you get the requied order. But I can't see how I can present this in the form of a solution. Help anyone?
• December 30th 2010, 09:24 AM
Drexel28
Quote:

Originally Posted by worc3247
Let A be an mxn matrix with entries in $\mathbb{R}$. Show by using elementary row and column operations, that there are invertible matrices P and Q (where P has size $m \cross m$ and Q has size $n \cross n$) such that:
$PAQ=$\left( \begin{array}{cc} I_r & 0 \\ 0 & 0 \\ \end{array} \right)$$.

I can sort of see why this is true. You would need to use ECO's to make all but 1 entries in a row 0, and then permute the rows until you get the requied order. But I can't see how I can present this in the form of a solution. Help anyone?

This isn't true. Take $m=n$ then the above would imply that $\displaystyle \det(A)=\frac{1}{\det(PQ)}\det\begin{pmatrix}I_r & 0\\ 0 &0\end{pmatrix}=0$ and so this isn't true if $A\in\text{GL}_n\left(\mathbb{R}\right)$. Right?
• December 30th 2010, 09:31 AM
FernandoRevilla
I suppose worc3247 meant "... and $\textrm{rank}(A)=r$ " then, the statement is true.

Fernando Revilla
• December 30th 2010, 09:32 AM
worc3247
I can see what you mean. Though I did write out the question excactly as it is on my problem sheet.
Nevertheless if someone could help me to explain why this is correct (assuming rank(A)=r) I would appreciate it.
• December 30th 2010, 09:43 AM
FernandoRevilla
Quote:

Originally Posted by worc3247
I can see what you mean. Though I did write out the question excactly as it is on my problem sheet.
Nevertheless if someone could help me to explain why this is correct (assuming rank(A)=r) I would appreciate it.