Linear Algebra Matrix Proof problem

[JKCB]

Homework Statement

Let A be an m x n matrix with rank m. Prove that there exists an n x m matrix B such that AB=Im

The Attempt at a Solution

I'm assuming I would need to start with the def. That there exists P an mxm invertible matrix and Q an nxn invertible matrix s.t. A=P(Im 0)Q

then P(Im 0)Q B = Im

now I might multiply left hand side by P inverse and right hand side by Q inverse.

I'm stuck am I going in the right direction?

 

[Office_Shredder]

I'm a bit confused by the notation here. What is I am 0? I'm assuming I am is the mxm identity matrix

 

[JKCB]

Yes I am is the mxm identity matrix and 0 is the mxn zero matrix.

I'm thinking of going another direction.

What if I start with letting B be any nxm matrix with rank n. Then AB would be an mxm matrix with rank m, then by the Thm (in my book) 2.18 corollary 2 an nxn matrix is invertible iff its rank is n. AB is invertible. Let M be the inverse then (AB)M=Im which means
A(BM)=Im therefore BM would be the nxm matrix we are looking for.

What do you think? Would that do it, any holes?

 

[Mark44]

Yes I am is the mxm identity matrix and 0 is the mxn zero matrix.

As Office_Shredder asked, what does I am 0 mean? Inquiring minds want to know.

 

[JKCB]

Im (I subscript m) is the mxm identity matrix and 0 is the m x (n-m) zero matrix.

 

[Mark44]

Then I_m 0 is the product of I_m and the m x (n - m) zero matrix, which is 0. Were you thinking that I_m times a zero matrix is something other than the same zero matrix?

 

[JKCB]

No. How about this? A= (Im 0)P where P is an nxm invertible matrix. Then replace A with (Im 0)P(B) = I am then (Im 0) P P^-1(Im)=Im
(0 ) (that is a column block matrix with I am being the Identity matrix and 0 being a zero matrix (n-m) x m) then that would make
B = p^-1 (Im)
(0 ) That is a partitioned matrix I am is mxm and the zero is (n-m) x m
Will that work?

 

[Office_Shredder]

(that is a column block matrix with I am being the Identity matrix and 0 being a zero matrix (n-m) x m

Is this what I am 0 is? It's hard to tell

 

[Mark44]

I think I have figured out what you're trying to communicate, but your notation was no help. What you are writing as (I_m 0) looks to me like a matrix product, and what you meant was the m x n matrix (I_m|0).