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Help with Linear Algebra T/F Questions

  1. Jun 7, 2012 #1
    Below is some statements for inverse, permutation and transposes. Next to them I will write what I believe to be correct. I know that at least one of my responses is incorrect, can anybody help me? I greatly appreciate it.

    Assume that all matrices are n\times n, that P and Q are permutation matrices and that R is a permutation matrix that interchanges two (otherwise unspecified) rows

    If A is symmetric and A=LU then L = U^T. F

    PQ=QP F

    R^-1 = R T

    R^15 = R F

    The inverse of an invertible symmetric matrix is symmetric. T

    A (square) matrix being invertible means the same as it being non-singular. T

    A nxn matrix is invertible if and only if elimination, possibly including row interchanges, produces n non-zero pivots. T

    The inverse of an invertible matrix is invertible. F

    (AB)^-1= (A^-1)(B^-1). F

    (AB)^-1= (B^-1)(A^-1). T

    if A and B are invertible, then so is AB. T
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 10, 2012 #2
    "The inverse of an invertible matrix is symmetric" T
    -This is not true. For example, consider the matrix [1,3\\1,2]]. Its inverse is [-2,3\\1,-1]], which is not symmetric.

    "The inverse of an invertible matrix is invertible" F
    -This actually is true. Let A be invertible, so A^-1=B is its inverse. Then B is certainly invertible, because BA=A^-1*A=I, and AB=A*A^-1=I. That is, A is the inverse of B.

    I'm assuming of course that you're working over a field.
     
  4. Jun 10, 2012 #3

    HallsofIvy

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    christoff, the statement was "The inverse of an invertible symmetric matrix is symmetric" and that is true.

    If R is a "permutation matrix that interchanges two (otherwise unspecified) rows" then it is true that [itex]R^{-1}= R[/itex]. And because that is true, it follows that [itex]R^2= R*R= R*R^{-1}= I[/itex]. From that it follows that R to any even power is I and R to any odd power is R.
     
  5. Jun 10, 2012 #4
    Ah, my apologies, and thank you for the clarification, HallsofIvy. I guess my eyes skipped over that word.
     
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