1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Prove: A 2x2 matrix is nonsingular if and only if the determinant != 0

  1. Aug 6, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove: A 2x2 matrix is nonsingular if and only if the determinant != 0

    3. The attempt at a solution

    I need to prove this, using logic and maybe the theorem that a n x n matrix is nonsingular if and only if it is row equivalent to I_n.

    I could use a push in the right direction. I need to show a proof for both ways, as it is an if and only if statement.
  2. jcsd
  3. Aug 6, 2010 #2
    I'm no linear algebra expert, but I would start by writing that if the matrix is written:
    a b
    c d

    Then it's singular if a = x*b and c=x*d (or a=x*c and b=x*d), and work my way to the formula for the determinant.
  4. Aug 6, 2010 #3
    Ok, but where does this come from?
  5. Aug 6, 2010 #4
    It would come from the fact that if the matrix is singular, then column one a multiple of the other (or one row is a multiple of the other). Maybe this isn't how it's supposed to be done though, so you might be better off waiting for somebody else's ideas than going off mine :/
  6. Aug 6, 2010 #5
    For one direction, assume matrix A is invertible. Then there is matrix B such that AB=1, where 1 is identity matrix. Take the determinant of both sides of AB=1.

    For the other direction, assume det A is invertible. Using the determinant and cofactor expansions, you can find a nice form for the inverse of A
  7. Aug 7, 2010 #6
    I probably should have mentioned. I don't have the definition of the determinant to work with.

    the problem actually says Show that the 2 x 2 matrix A is nonsingular if and only if ad-bc != 0.

    I've figured out the If matrix A is nonsingular, then ab-bc != 0 side. I just need the if ad - bc != 0, Matrix A is nonsingular side.
    Last edited: Aug 7, 2010
  8. Aug 8, 2010 #7
    Given matrices of the form
    a b = A
    c d


    d -b = B
    -c a

    How can you ensure AB=1?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook