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Math Help - linear algebra urgent help please

  1. #1
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    Question linear algebra urgent help please

    true or false and why?
    1) every square matrix can be written as the product of elementary matrices.
    2) The inverse of an elementary matrix is an elementary matrix.
    3) Every system with 3 equations in 4 unknowns has at least one solution.

    4) A square matrix is defined to be stochastic if all its elements are nonnegative and the sum of the elements in any given column is 1.
    Prove that the product of any two stochastic matrices is stochastic.

    please help me urgently. Thank you very much.
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  2. #2
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    Quote Originally Posted by happystudent View Post
    true or false and why?
    1) every square matrix can be written as the product of elementary matrices. Mr F says: False since the statement is not true when det(M) = 0.

    2) The inverse of an elementary matrix is an elementary matrix. Mr F says:
    True.

    3) Every system with 3 equations in 4 unknowns has at least one solution. Mr F says:
    False. It's not hard to come up with a counter-example to the statement.

    4) A square matrix is defined to be stochastic if all its elements are nonnegative and the sum of the elements in any given column is 1.
    Prove that the product of any two stochastic matrices is stochastic.
    [snip]
    Where are you stuck with 4)?
    Last edited by mr fantastic; August 2nd 2008 at 09:06 PM.
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  3. #3
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    i don't understand 4. so i don't know what to start
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  4. #4
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    Quote Originally Posted by happystudent View Post
    [snip]
    4) A square matrix is defined to be stochastic if all its elements are nonnegative and the sum of the elements in any given column is 1.
    Prove that the product of any two stochastic matrices is stochastic.
    [snip]
    Quote Originally Posted by mr fantastic View Post
    Where are you stuck with 4)?
    Quote Originally Posted by happystudent View Post
    i don't understand 4. so i don't know what to start
    Let C = AB where A and B are both nxn stochastic matrices (column).

    By definition:

    Sum of entries in rth column of A: \sum_{i=1}^{n} a_{ir} = 1

    Sum of entries in rth column of B: \sum_{i=1}^{n} b_{ir} = 1

    By definition of matrix multiplication:

    Element of C in ith row and jth column: c_{ij} = \sum_{r=1}^{n} a_{ir} b_{rj}.

    Therefore the sum of the entries in the jth column of C is \sum_{i=1}^{n} c_{ij} = \sum_{i=1}^{n} \sum_{r=1}^{n} a_{ir} b_{rj}

    = \sum_{r=1}^{n} \left( \sum_{i=1}^{n}  a_{ir} \right) b_{rj} = \sum_{r=1}^{n} (1) b_{rj} = (1) (1) = 1

    where the justification of the last line is left for you.
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