- #1
Buffu
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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##.