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Math Help - Matrix inverses, determinants... questions

  1. #1
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    Matrix inverses, determinants... questions

    1. If A is a square matrix satisfying A^2 +10A = I, where I is the identity matrix with the same dimensions as A, does A have an inverse?

    I know the answer is yes, because I found the inverse, 10I + A, basically by trial and error.

    But is there a more elegant/systematic way to go about this in general?

    2. If A is a square matrix with determinant d, & \lambda is a number, what is the determinant of \lambda A?

    I know the answer, but I don't know why it is..


    Thanks for any help
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  2. #2
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    Quote Originally Posted by Aileys. View Post
    1. If A is a square matrix satisfying A^2 +10A = I, where I is the identity matrix with the same dimensions as A, does A have an inverse?

    I know the answer is yes, because I found the inverse, 10I + A, basically by trial and error.

    But is there a more elegant/systematic way to go about this in general?

    2. If A is a square matrix with determinant d, & \lambda is a number, what is the determinant of \lambda A?

    I know the answer, but I don't know why it is..


    Thanks for any help
    1. A^2 +10A = I \Rightarrow A^{-1} A^2 +10 A^{-1} A = A^{-1} I \Rightarrow A + 10 I = A^{-1}.

    A thread of related interest: http://www.mathhelpforum.com/math-he...-inverses.html

    2. Do you know what happens to \det(A) when you multiply a row of A by \lambda? Now note that every row of A is multiplied by \lambda ....
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  3. #3
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    Using the Cayley-Hamilton theorem you can use the expression in the first part of you question to find the characteristic polynomial. Now seeing as how it is only a quadratic you should have no problem finding the root. So from here you should be able to know the determinant and if it is in fact invertible.
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  4. #4
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    Quote Originally Posted by Aileys. View Post
    1. If A is a square matrix satisfying A^2 +10A = I, where I is the identity matrix with the same dimensions as A, does A have an inverse?

    I know the answer is yes, because I found the inverse, 10I + A, basically by trial and error.
    But is there a more elegant/systematic way to go about this in general?
    You don't need trial and error: A^2+ 10A= A(A+ 10I)= (A+ 10I)A= I immediately tells you that A+ 10I is the inverse of A.


    Quote Originally Posted by Aileys. View Post
    2. If A is a square matrix with determinant d, & \lambda is a number, what is the determinant of \lambda A?

    I know the answer, but I don't know why it is..

    Thanks for any help
    Think about the special case in which A is a diagonal matrix.
    Last edited by mr fantastic; August 10th 2009 at 11:20 PM. Reason: Fixed a bit of latex, added an open tag for the second quote.
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  5. #5
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    Quote Originally Posted by Aileys. View Post

    2. If A is a square matrix with determinant d, & \lambda is a number, what is the determinant of \lambda A?

    I know the answer, but I don't know why it is..
    The determinant of a n\times n square matrix can be written as the sum of products of n distinct terms at a time multiplied by + or - 1 (which elements appear in the products does not matter for the purposes of answering the question) now as each element is multiplied by \lambda the determinant is multiplied by \lambda^n.

    CB
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