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Math Help - Some problems

  1. #1
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    Oct 2005
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    Some problems

    I have a ton of homework, but I am stuck on 4 of them.

    1. Find non-invertible matrices A and B, such that AB is invertible.


    2. Find, with proof, a square matrix A with entries in Z2 such A is not the sum of a symmetric and anti-symmetric matrix.


    3. A square matrix X is called idempotent if X^2 = X. Prove that if AB = A and BA = B then A and B are idempotent.


    4. Let Pn(x) be the vector space of all polynomials of degree <= n with real number coefficients. Prove that the set
    {1, x, x(x − 1), x(x − 1)(x − 2), . . . , x(x − 1)(x − 2) (x − n + 1)}
    is a basis for this vector space.

    If anybody can help me with any I will appreciate it.
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  2. #2
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    For (1) note that it isn't true for square matrices, so you have to think of non-square A and B.

    For (2), note that over Z/2 an antisymmetric matrix is also symmetric and that the sum of symmetric matrices is symmetric.

    For (3) write A^2 = (AB)(AB) = A(BA)B = ABB = (AB)B = AB = A.

    For (4) consider the degrees.
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  3. #3
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    Oct 2005
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    thank you,
    I will try them and if I have any other problems I will let you know.
    thank you again
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