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: Square matrix and its transpose satisfying an equation

  1. Mar 25, 2015 #1
    1. The problem statement, all variables and given/known data
    Show that if a square matrix A satisfies

    A3 + 4A2 -2A + 7I = 0
    Mod note: It took me a little while to realize that the last term on the left is 7I, seven times the identity matrix. The italicized I character without serifs appeared to me to be the slash character /.

    then so does AT

    2. Relevant equations

    3. The attempt at a solution
    What I notice is that for any n x n matrix A and powers thereof, the diagonals of A and the transpose are the same. I experimented with a 2 x 2 matrix (with entries a, b, c, d), squared and cubed to see what happens and the result, aside from being somewhat messy, ends with each matrix reducing to I when appropriate row operations are applied. I'm not sure how to proceed from here or if I'm even on the right track with this thinking. Please assist.
    Last edited by a moderator: Mar 25, 2015
  2. jcsd
  3. Mar 25, 2015 #2


    User Avatar
    Science Advisor
    Homework Helper

    You are thinking about this too hard. Just take the transpose of both sides of that equation. For example, what is ##(A^3)^T## in terms of ##A^T##.
  4. Mar 25, 2015 #3
    Damn. I really should have seen that, unbelievably simple. Of course , hindsight is always 20/20. Thanks.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted