Linear Algebra: The vector space R and Rank

[rad0786]

Two m x n matrices A and B are called EQUIVALENT (writen A ~e B if there exist invertible matracies U and V (sizes m x m and n x n) such that A = UBV
a) prove the following properties of equivalnce
i) A ~e A for all m x n matracies A
ii) If A ~e B, then B ~e A
iii) A ~e B and B~e C, then A~e C

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Lets just do part i) for now...

It seems ovicus that "A ~e A for all m x n matracies A"... but.. here's how i would do it

A = m x n
U = m x m
V = n x n

So

A = UAV
A = (UA)V UA = (m X m)(m X n) = (m X n)
A = (UA)(V) UAV = (m X n)(n x n) = (m x n)
A = A

How does that sound? Is that how you prove this? Or do i have the wrong idea?

Please help

thanks

 

[matt grime]

What is your idea? How can A=m x n? That makes no sense. A is an mxn matrix, but not equal to m x n, whatever that means.
Find U and V, invertible matrices such that A=UAV, that's all you need to do.
You can't start by setting A=UAV, as you do, since that is assuming the answer. I don't even know what your argument is trying to do since you haven't used any words to explain any of your steps.
As for the other two: if you actually write out what you need to show, and what you are given, then it is a simple manipulation of matrices.