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Are these statements TRUE or FALSE and why?

  1. Mar 3, 2009 #1
    Hey,

    Just stumbled into some linear algebra questions in my textbook...that I can't quite seem to work out...

    Prove or disprove the following statements concerning 2 x 2 matrices:

    1) If A^2 - 3A- 2I = 0 then (A-1) and (A-2I) are both invertible.

    (so I got the determinant to be 0, which would mean that they are not, thus making it FALSE..is this right?)

    2) If A = EB and E is elementary then B = FA for some elementary F.

    ...I wasn't too sure about this one...

    Thank you...Neon Vomitt was the one who sparked my interest...so THANK YOU to you!
     
  2. jcsd
  3. Mar 3, 2009 #2
    Each elementary matrix represents a linear operation that can be undone. Add some rigor for a direct proof.
     
  4. Mar 4, 2009 #3
    Regarding the first question. To be able to factor the left side you must have +2I, so you have to add 4I to both sides to get (A-I)(A-2I)=4I.
    If you then take the determinant you get det(A-I)det(A-2I)=4, so (A-I) and (A-2I) must be invertible.
     
  5. Mar 4, 2009 #4

    HallsofIvy

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    Do you think the fact that A^2- 3A- 2I= (A- I)(A- 2I)= 0 is important?

    Every elementary matrix corresponds to some row operation and every row operation has a specific "inverse row operation".

    The row operations are:
    1) Swap two rows. And its inverse is itself: "Swap the same two rows"
    2) Multiply a row by the number a. And its inverse is "Multiply the same row by 1/a".
    3) Add a times a one row to a second row. And its inverse is "Subtract a times the same row from that row."

    And, in fact, your statement isn't quite true because of a problem with (2) above. "multiply and entire row by 0" is a row operation although not used very often for obvious reasons! The elementary matrix corresponding to that row operation is the identity matrix with one row changed to all 0s. If E is such a matrix, there is no such F.

     
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