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Math Help - [SOLVED] Linear Algebra- Diagonalizability

  1. #1
    kutcha
    Guest

    [SOLVED] Linear Algebra- Diagonalizability

    PLEASE HELP!!!

    a) Let A=
    1 0 0
    0 2 0
    0 0 3

    Suppose that B is a 3x3 matrix and that AB=BA. Show that B is also a diagonal 3x3 Matrix

    b) Suppose that A is an 3x3 real matrix with three distinct eigenvalues. Let B be another 3x3 matrix for which AB=BA. Show that B is diagonalizable.

    c) Suppose that A and B are diagonalizable matrices with precisely the same eigenspaces (but not necessarily the same eigenvalues). Prove that AB=BA


    THANKS!
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  2. #2
    MHF Contributor kalagota's Avatar
    Joined
    Oct 2007
    From
    Taguig City, Philippines
    Posts
    1,026
    Quote Originally Posted by kutcha View Post
    PLEASE HELP!!!

    a) Let A=
    1 0 0
    0 2 0
    0 0 3

    Suppose that B is a 3x3 matrix and that AB=BA. Show that B is also a diagonal 3x3 Matrix
    If you suppose B to be an arbitrary matrix, say
    B =
    a b c
    d e f
    g h i

    and perform the matrix multiplication on both sides, you should come up and be forced that the values of b,c,d,f,g,h is 0..
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