1. ## Matrix Arithmetic Help...

I am having two problems here. The first one I am somewhat confident of:

1. Show that in general, for any square matrix satisfying:
$\displaystyle A^2 - 2A + 5I = 0$
then
$\displaystyle A^{-1} = \frac{1}{2}(2 - A)$

I proceeded as follows:
$\displaystyle A^2A^{-1} - 2AA^{-1} + 5IA^{-1} = 0A^{-1}$
$\displaystyle A - 2I + 5A^{-1} = 0$
$\displaystyle A^{-1} = \frac{1}{2}(2 - A)$

While what I did seems legitimate...how do I show that the matrices must be square?

2. Let A, D, and P be n X n matrices satisfying $\displaystyle P^{-1}AP = D$. Solve for A. Must it be true that A = D?

For this one I am not sure how to proceed. In order to get rid of P I need to say multiply by $\displaystyle P^{-1}$ but that doesn't seem to help me... Same sort of situation for P.

2. Originally Posted by alterah
i am having two problems here. The first one i am somewhat confident of:

1. Show that in general, for any square matrix satisfying:
$\displaystyle a^2 - 2a + 5i = 0$
then
$\displaystyle a^{-1} = \frac{1}{2}(2 - a)$

i proceeded as follows:
$\displaystyle a^2a^{-1} - 2aa^{-1} + 5ia^{-1} = 0a^{-1}$
$\displaystyle a - 2i + 5a^{-1} = 0$
$\displaystyle a^{-1} = \frac{1}{2}(2 - a)$

while what i did seems legitimate...how do i show that the matrices must be square? mr f says: Does a non-square matrix have an inverse ....?

2. Let a, d, and p be n x n matrices satisfying $\displaystyle p^{-1}ap = d$. Solve for a. Must it be true that a = d?

For this one i am not sure how to proceed. In order to get rid of p i need to say multiply by $\displaystyle p^{-1}$ but that doesn't seem to help me... Same sort of situation for p.
2. $\displaystyle p p^{-1}ap p^{-1} = p d p^{-1}$ ....

Edit: That annoying intermittent glitch that changes any posted upper case letter into a lower case letter appears to have struck again.

3. Wow...there are definitely those moments in math when you are like "How did I not simply see that I made the problem harder than it was. What is your input on the first problem? Thanks!

4. Originally Posted by Alterah
[snip]
What is your input on the first problem? Thanks!
I have already given my input (in the form of a question for you to ponder).

5. Whoops...didn't see the text...in red in your quote.