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Show that if A is nonsingular symmetric matrix

  1. Dec 2, 2003 #1
    Hello everyone! Can anyone help me here in this theorems (prove)?
    (Or solve)

    1. Suppose that A and B are square matrices and AB = 0. (as in zero matrix) If B is nonsingular, find A.

    2. Show that if A is nonsingular symmetric matrix, then A^-1
    is symmetric.

    I hope these won't bother...
  2. jcsd
  3. Dec 2, 2003 #2


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    For the first one, study the definition your book or notes gives for "nonsingular". Look at it again. How does that definition bear of the product AB = 0?

    For the second one, do they give you a method for finding the inverse of a square matrix? What happens in that method when aij = aji for all i and j?
  4. Dec 4, 2003 #3
    Thank you


    I got the answer... for the first, A must be a zero matrix. But
    there are times when a matrix multiplied by a nonzero matrix can still result to a product of zero matrix. When is that?

    For no. 2, I got it already. it's a matter of direct substitution using the properties given in my lecture.
  5. Dec 4, 2003 #4


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    Well, you know that a non-zero matrix multiplied by the zero matrix gives the zero matrix so I assume you mean "a nonzero matrix multiplied by a non-zero matrix" can give a zero matrix.

    You have just determined that if one of the matrices is non-singular, then the other must be the zero matrix. What happens if you multipy two singular matrices?

    I think you will find that true for many problems! Amazing that a lecture (and text book) can actually be useful isn't it?
  6. Dec 5, 2003 #5
    I got it


    You mean that the product of 2 singular (square ) matrices must result to either another matrix or a zero matrix?

    Oh, thank you.
  7. Dec 6, 2003 #6


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    "either another matrix or the zero matrix"???

    Well that's always true isn't it! :smile:

    What I meant was that, since you have already proved that if AB= O with A non-singular then B=0, the ONLY way you could get AB= 0 without either A or B 0 is to multiply two singular matrices.
    The product of two singular matrices is not always 0 but is always a singular matrix.
  8. Dec 12, 2003 #7

    Hi again.

    Thank you very much! =)
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