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Linear Algebra problem

  1. Feb 20, 2008 #1
    The problem is prove that for every square singular matrix A there is a nonzero matrix B, such that AB equals the zero matrix.

    I got AB to equal the idenity matrix, but have no clue how to get it to the zero matrix.
     
  2. jcsd
  3. Feb 20, 2008 #2
    If A is viewed as the matrix of a linear transformation, then it being singular is the same as saying that it maps a subspace in the domain to the 0 subspace of the range. Find the kernel of A and let B be a matrix of vectors in that space.
     
  4. Feb 21, 2008 #3
    you said:
    I got AB to equal the idenity matrix, but have no clue how to get it to the zero matrix.

    how did you do that? If A is singular then it doesn't have an inverse, but you found one namely B.
     
  5. Feb 21, 2008 #4

    HallsofIvy

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    If A is singular, det(A)= 0. Also det(AB)= det(A)det(B). You, apparently, have proved that det(AB)= 0(det(B)= 0= det(I)= 1. A miracle indeed!
     
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