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Matrix similarity

  1. Jan 7, 2008 #1
    Question 1
    If A and B are similar matrices, then prove that A is nonsingular if and only if B is nonsingular.

    MY SOLUTION:
    B is nonsingular if it's columns are linearly independent
    P[tex]^{-1}[/tex]AP=B where the main diagonal of B is made of eigenvectors of A.
    How does this affect the nonsingularity of A ????? this is where I am stuck.

    Question 2
    If A and B are similar matrices, Show that if B=PAP[tex]^{-1}[/tex] then det(B)=det(A)

    MY SOLUTION
    As B=PAP[tex]^{-1}[/tex] this implies that BP=PA
    Taking the determinant of both sides.
    det(BP)=det(PA)
    when expanded, we have det(B).det(P)=det(P)det(A) therefore det(B)=det(A)
    1. The problem statement, all variables and given/known data



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    3. The attempt at a solution
     
  2. jcsd
  3. Jan 7, 2008 #2

    Defennder

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    Homework Helper

    1. Here's an easier way to think about it: Let A, B be square matrices of same size. AB = C. Note that [tex]C^{-1} = B^{-1}A^{-1}[/tex]. Which effectively means that if C is invertible (non-singular), then it must be the case that A,B whose matrix product make up C must also be invertible. You can generalise this to mean that if a square matrix A is invertible, then its "product components" (ie. all the possible square matrices which maybe multiplied together to give A) must also be invertible.

    2. I think your answer is ok.
     
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