Results 1 to 2 of 2

Math Help - Determinantal rank

  1. #1
    Member
    Joined
    Dec 2010
    Posts
    107

    Determinantal rank

    Let A be an n n matrix. Define the determinantal rank detrankA to be the greatest value of r such that there is an r r submatrix B of A (that is, a matrix B obtained by deleting n − r rows and n − r columns from A) such that detB \neq 0. Show that detrankA = rankA.

    Genuinally have no idea where to start, I can't even see that this is true so haven't got any ideas. Help appreciated!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Nov 2010
    Posts
    193
    The rank of a matrix is equal to the maximal number of linearly independent rows (and the same number of columns). If you delete the rest of the rows and columns, keeping this set of linearly independent ones, what can we say about the determinant of the matrix you have left over? (Consider what the reduced row-echelon form of the new matrix must look like, for instance. Or consider what the determinant says about invertibility of the matrix.)

    Conversely, if you have a submatrix with nonzero determinant, can we say whether or not its rows (columns) must be linearly independent or not?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proof: rank(AB)+n >= rank(A)+rank(B)
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: September 9th 2010, 05:28 PM
  2. Replies: 3
    Last Post: August 20th 2010, 05:32 AM
  3. Rank(T) = Rank(L_A) proof?
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: July 17th 2010, 08:22 PM
  4. determinantal rank
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: March 10th 2010, 08:39 AM
  5. Short proof that rows-rank=column-rank?
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: June 26th 2009, 10:02 AM

Search Tags


/mathhelpforum @mathhelpforum