Self Study problem help

This is a problem from the third edition of 'Abstract Algebra' by David
S. Dummit and Richard M. Foote. I am studying these problems on my own,
but would apprechiate only minimal feedback to help me solve it on my
own, instead of an outright solution.

The problem goes goes like this...

Given a 2X2 matrix M with first row (1,1) and second row (0,1) we
consider all the real valued 2X2 matrices X such that XM = MX and we'll
call this set B. The question asks given a real valued 2X2 matrix with
first row (p,q) and second row (r,s) what conditions on p,q,r and s
presisely when this matrix is an element of B.

Now the work I've done is to take the p,q,r,s matrix and multiply it on
the left and right sides by the matrix M and then compare the
components of each matrix to see how it restricts those elements. I
found that p = p+r which implied that r=0 and that p+q = q+s which
implied that p=s. This led me to believe that each matrix must have
first row (p,q) and second row (0,p).

Now my question is simple - did I do this correctly? I have checked
that matrices of this for by plugging through a few examples, I'm just
not entirely convinced that I have found the complete restrictions on
p,q,r,s or if I'm expressing this in the correct way. Moreso, I'm
unsure if my conditions are too sever and I've somehow excluded some
allowable matrices which don't fit this form.

I also had one more question which I am fairly confident that I
answered correctly, but would simply like a confirmation so that I know
I'm on the right track or if I've got the wrong idea. The question is
from the same text, and reads...

Determine wether the function f: from R+ to Z defined by mapping a real
number r to the first digit to the right of the decimal point in a
decimal expansion of r is well defined. I said that it wasn't well
defined since 0.999~ and 1.000~ represent the same real number yet
would map to 9 and 0 respectively.

Thanks in advance, Abe.

.

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