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Homework Help: Is an invertible matrix always lin. independent?

  1. May 14, 2009 #1
    Because according to my book,

    If A's an invertible nxn matrix, then the eq. Ax=b is consistent for EACH b...

    this consistency would imply lin. independency?
  2. jcsd
  3. May 14, 2009 #2


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    Gold Member

    Yes. Also, the determinant of an invertible matrix is non-zero. The determinant of a matrix of linearly dependent vectors is 0.
  4. May 14, 2009 #3


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    Science Advisor

    Strictly speaking, the term "independent" applies to sets of vectors, not matrices so it does not make sense to ask if a matrix is independent. What you are really asking is whether the rows or columns of the matrix, thought of as vectors are independent. The answer to that is yes. If a matrix is invertible, then its rows (or columns), thought of as a set of vectors, is indepependent.
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