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Math Help - a problem regarding rank of matrices

  1. #1
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    a problem regarding rank of matrices

    Hey guys,
    can anyone help me with this problem please ??

    Given A be m * n (m by n) matrix and B be n * p (n by p) matrix

    Show that 1. rank(AB) is less than or equal to rank(A)
    2. rank(AB) is less than or equal to rank(B)
    3. If B is invertible, than rank(AB) = rank(A)

    i think 1 and 2 are kind of similar ? so clear explanation on 1 and 3 are appreciated. Thanks so much!!
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  2. #2
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    Quote Originally Posted by dull1234 View Post
    Hey guys,
    can anyone help me with this problem please ??

    Given A be m * n (m by n) matrix and B be n * p (n by p) matrix

    Show that 1. rank(AB) is less than or equal to rank(A)
    2. rank(AB) is less than or equal to rank(B)
    3. If B is invertible, than rank(AB) = rank(A)

    i think 1 and 2 are kind of similar ? so clear explanation on 1 and 3 are appreciated. Thanks so much!!
    The rank of A can at most be m, the rank of B can at most be n.

    AB=C_{m x p}; therefore, the rank of C can at most be m.

    Do we know if n\leq m?
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