Prove that a matrix can be reduced to RRE and CRE

[Buffu]

Homework Statement

Let $A$ be an $m \times n$ matrix. Show that by means of a finite number of elementary row/column operations $A$ can be reduced to both "row reduced echelon" and "column reduced echelon" matrix $R$. i.e $R_{ij} = 0$ if $i \ne j$, $R_{ii} = 1$, $1 \le i \le r$, $R_{ii} = 0$ if $i > r$. Also show that $R = PAQ$ where $P$ and $Q$ are invertible $m\times m$ and $n \times n$ matrices respectively.

Homework Equations

The Attempt at a Solution

Since I know I can pass $A$ to a row reduced echelon matrix in finite number of operations.
Lets say the row reduced echelon form of $A$ is $R^{\prime}$. Then $R^\prime = PA$.

Also since nothing is special about rows, therefore I can say that a matrix can be passed on to column reduced echelon in finite number of steps. Therefore I can pass $R^\prime$ to a column reduced form $R$ in finite number of steps. Let's say $R = QR^\prime$

From above I can say $A$ can be passed to a column and row reduced echelon form in finite number of steps and $R = QPA$.

Is this correct ? I think it is wrong since I used a lot of words and also I got $R = QPA$ not $R = PAQ$.

 

[Ray Vickson]

Homework Statement

Let $A$ be an $m \times n$ matrix. Show that by means of a finite number of elementary row/column operations $A$ can be reduced to both "row reduced echelon" and "column reduced echelon" matrix $R$. i.e $R_{ij} = 0$ if $i \ne j$, $R_{ii} = 1$, $1 \le i \le r$, $R_{ii} = 0$ if $i > r$. Also show that $R = PAQ$ where $P$ and $Q$ are invertible $m\times m$ and $n \times n$ matrices respectively.

Homework Equations

The Attempt at a Solution

Since I know I can pass $A$ to a row reduced echelon matrix in finite number of operations.
Lets say the row reduced echelon form of $A$ is $R^{\prime}$. Then $R^\prime = PA$.

Also since nothing is special about rows, therefore I can say that a matrix can be passed on to column reduced echelon in finite number of steps. Therefore I can pass $R^\prime$ to a column reduced form $R$ in finite number of steps. Let's say $R = QR^\prime$

From above I can say $A$ can be passed to a column and row reduced echelon form in finite number of steps and $R = QPA$.

Is this correct ? I think it is wrong since I used a lot of words and also I got $R = QPA$ not $R = PAQ$.

So, you can have $R = QPA,$ and this can be written as $R = P_1 A Q_1$, where $P_1 = QP$ and $Q_1 = I$ (the $n \times n$ identity matrix).

 

[Buffu]

So, you can have $R = QPA,$ and this can be written as $R = P_1 A Q_1$, where $P_1 = QP$ and $Q_1 = I$ (the $n \times n$ identity matrix).

Then the proof is correct ?

 

[Ray Vickson]

Then the proof is correct ?

What proof? All you did was make statements; you did not really "prove" anything.

 

[Buffu]

What proof? All you did was make statements; you did not really "prove" anything.

No I did not get what you are saying. Isn't statements like " You can pass from from A to a row/column reduced form in finite steps " is proven to be true.
So I just need to combine these types of statements to form a proof.