16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 1.6.11

Systems of Linear Equations

Mathematics — Linear Algebra · row-column-reduction

Problem

Let

A

be an

m \times n

matrix. Show that by means of a finite number of elementary row and/or column operations one can pass from it to a matrix

R

which is both 'row-reduced echelon' and 'column-reduced echelon,' i.e.,

R_{ij} = 0

i \neq j

R_{ii} = 1

1 \leq i \leq r

R_{ii} = 0

i > r

. Show that

R = PAQ

, where

P

is an invertible

m \times m

matrix and

Q

is an invertible

n \times n

matrix.