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Linear Algebra.

  1. Nov 16, 2008 #1
    1. The problem statement, all variables and given/known data

    Show that if AAt(where At is the transpose of A) has no inverse,Then A itself cannot have an inverse

    2. The attempt at a solution

    Here's what i did


    thus AAt has an inverse IF and only IF A is invertible and since AAT has no inverse then A has no inverse ...

    i thinkt it's somehow wrong the teacher told me that i assumed that AAt is invertible but it's given without an inverse ..any suggestions would be appreciated :)
  2. jcsd
  3. Nov 16, 2008 #2


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    Staff Emeritus
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    You've assumed it's invertible by writing down (AAT)-1

    Hint: Consider determinants...
  4. Nov 16, 2008 #3
    i haven't taken determinant till now.. is there any other way to do it ?
  5. Nov 16, 2008 #4


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    Staff Emeritus
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    You want to prove "If AAT does NOT have an inverse, then A itself does NOT have an inverse". With all those "not"s, that's what I would call a "negative" statement. It is almost easier to prove the contraposative of a "negative" statement than the statement itself because then you can make a "positive" statement.

    Here the contrapositive is "If A has an inverse then so does AAT."
  6. Nov 16, 2008 #5
    you mean i should assume that A is invertible and prove that AAT is invertible also ...
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