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!

Matrix determinants and inverses

  1. Mar 2, 2009 #1
    1. The problem statement, all variables and given/known data
    If A and S are n x n matrices with S invertible, show that det(S-1AS)=det(A). [HINT: Since S-1S=In, how are det(S-1) and det(S) related?]

    2. Relevant equations

    3. The attempt at a solution

    Not sure. The only thing I can think of doing is substituting S-1S=In into det(S-1AS), so you have det(InA)=det(A), but I really don't know. Can somebody get me started so we can work through it?
  2. jcsd
  3. Mar 2, 2009 #2


    User Avatar
    Science Advisor
    Homework Helper

    Sure. You should have proved det(AB)=det(A)*det(B). det(I)=1. Start from there.
  4. Mar 3, 2009 #3
  5. Mar 3, 2009 #4


    User Avatar
    Science Advisor

    You don't seem to have payed any attention to the hint! "S-1S=In, how are det(S-1) and det(S) related?" I presume you know that det(AB)= det(A)det(B).
  6. Mar 3, 2009 #5
    Is it:

    det(S-1AS) = det(S-1S)det(A)




  7. Mar 3, 2009 #6


    User Avatar
    Science Advisor
    Homework Helper

    That's true. But don't make it look like you think the matrices commute. S^(-1)AS is NOT necessarily equal to S^(-1)SA. But det(S^(-1)AS)=det(S^(-1))*det(A)*det(S). Now you can rearrange.
  8. Mar 3, 2009 #7
    So first put them in 3 seperate determinants like you did, then move the S-1 next to S, then put it back together? Go backwards from two determinants multiplying each other to a single determinant, which would be equal to det(S-1S) ?
  9. Mar 3, 2009 #8


    User Avatar
    Science Advisor
    Homework Helper

    That's it exactly.
  10. Mar 3, 2009 #9
    Thanks for the help :smile:

    Just on a curious note, are there examples of when doing what I did in post #5 would be wrong, or is it just an incorrect way of solving?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Matrix determinants and inverses
  1. Inverse matrix (Replies: 3)

  2. Matrix inversion (Replies: 6)

  3. Inverse of a Matrix (Replies: 2)

  4. Matrix inverse? (Replies: 2)

  5. Inverse matrix (Replies: 6)