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Math Help - Matrix Arithmetic Help...

  1. #1
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    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:
     A^2 - 2A + 5I = 0
    then
    A^{-1} = \frac{1}{2}(2 - A)

    I proceeded as follows:
     A^2A^{-1} - 2AA^{-1} + 5IA^{-1} = 0A^{-1}
    A - 2I + 5A^{-1} = 0
    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 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 P^{-1} but that doesn't seem to help me... Same sort of situation for P.
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  2. #2
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    Quote Originally Posted by alterah View Post
    i am having two problems here. The first one i am somewhat confident of:

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

    i proceeded as follows:
     a^2a^{-1} - 2aa^{-1} + 5ia^{-1} = 0a^{-1}
    a - 2i + 5a^{-1} = 0
    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 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 p^{-1} but that doesn't seem to help me... Same sort of situation for p.
    2. 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.
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  3. #3
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    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!
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  4. #4
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    Quote Originally Posted by Alterah View Post
    [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).
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  5. #5
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    Whoops...didn't see the text...in red in your quote.
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