1. Not finding help here? Sign up for a free 30min 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!

Demonstrate that a matrix that has a null row is not invertible

  1. Jan 19, 2009 #1

    fluidistic

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    Demonstrate that a matrix that has a null row is not invertible.

    2. The attempt at a solution I know that saying that a matrix A is invertible is equivalent to say that A is row-equivalent and column-equivalent to the identity matrix.
    And also that there exist elemental matrices E_1, E_2,...,E_r and F_1,...,F_s such that the identity matrix is equal to E_1...E_r A F_1...F_s.
    So I could show that if A has at least a null row, then one of these properties cannot be true.
    But still I don't know how to proceed. I understand clearly that if A has a null row, it's obvious that it cannot be equivalent to the identity matrix... but to prove it formally, I don't know how I can start. A little help is welcome.
     
  2. jcsd
  3. Jan 19, 2009 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Suppose it did have an inverse. What does its product with its inverse supposed to look like? How does it actually look?
     
  4. Jan 19, 2009 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Think of A as the coefficient matrix such that Ax= b and let xn be the component of x corresponding to the 0 row. If the corresponding component of b is not 0, there is NO value of xn that makes that true. If the corresponding component of b is 0, then any value of xn makes that true. In either case there is NOT single solution to Ax= b. If A-1 exists, then A-1b would be 'the' solution to Ax= b.
     
  5. Jan 19, 2009 #4

    fluidistic

    User Avatar
    Gold Member

    It is supposed to look like the identity matrix, however if the matrix has a null row it means that the product of this matrix with its inverse has a null column which means it's not the identity matrix, so that in fact any matrix having a null row doesn't have an inverse, hence is not invertible.
    Is it a good proof? Or too informal?
     
  6. Jan 19, 2009 #5

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I can't answer that one; it would depend on what your teacher expects.
     
  7. Jan 20, 2009 #6

    fluidistic

    User Avatar
    Gold Member

    Ok I understand. I hope it's enough.
    And thank you very much HallsOfIvy, I got what you mean... nice way to do the proof.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?