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Can the basis minor of a matrix be the matrix itself?

  1. Oct 19, 2015 #1
    Hello

    I am trying to learn linear algebra, and I came across this definition of basis minor on this webpage:

    https://en.wikibooks.org/wiki/Linear_Algebra/Linear_Dependence_of_Columns

    "The rank of a matrix is the maximum order of a minor that does not equal 0. The minor of a matrix with the order of the rank of the matrix is called a basis minor of the matrix, and the columns that the minor includes are called the basis columns."

    Does this mean that if the determinant of the matrix does not equal zero, then its basis minor is just itself, and its rank is just the same as its order?

    Thank you
     
  2. jcsd
  3. Oct 19, 2015 #2

    Geofleur

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    That sounds right. For an ## n \times n ## matrix, the order of a minor is allowed to be the same as ## n ##. If the determinant of the whole matrix does not vanish, then rank is ## n ##, and the order of the largest non-vanishing minor is also ## n ##.
     
  4. Oct 19, 2015 #3
    Thanks Geofleur. I thought so but just wasn't sure.
     
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