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Some T/F on matrices

  1. Apr 30, 2007 #1
    1. If B is formed from A by exchanging two rows then B is similar to A
    2. If a triangular matrix is similar to diagonal, it is already diagonal
    3. If A and B are diagonalizable so is AB
    4. Every invertible matrix can be diagonalized

    My answers:
    1. F: eigenvalues change, so the Jordan form would be different
    2. I want to say true, but not entirely sure: eigenvalues would be the same and the jordan form is same so i guess that the triangular would be diagonal?
    3. i would guess true, but not sure how to motivate
    4. F, I know that for sure because we had an example in class
    [1 -1]
    [0 1] which is invertible but not diagonalizable since it has only 1 eigenvector, but how would I motivate it theoretically, i.e. how would I say in general that in certain cases there are not enough eigenvectors even for invertible matrices => not diagonalizable?

  2. jcsd
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