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!

Matrix Equations

  1. Mar 8, 2016 #1
    1. The problem statement, all variables and given/known data
    ef19e041d3.png

    2. Relevant equations
    Inverse of an (nxn) (n=2 only) square matrix:
    2d95cc5b65.png
    3. The attempt at a solution

    c318f161da.jpg

    The answer provided in the solutions does the exact same thing except, where my ?? are. It does A = BCB^-1. Where as I do A = CBB^-1. When I was doing this question I was wondering the same thing. I know matrix multiplication isnt associative( AB isnt equal to BA), so how do I know which way to form the equations? I mean how am I meant to know whether its meant to be A=CBB^-1 or BCB^-1?
     
  2. jcsd
  3. Mar 8, 2016 #2

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    Matrix multiplication is associative. You probably meant to write that matrix multiplication isn't commutative.

    Since matrix multiplication isn't commutative, when you multiply one side of an equation to the left, you must multiply the other side of the equation to the left as well. Same for multiplying to the right.
    So if ##X=Y##, then ##AX=AY## and ##XA=YA##, but not necessarily ##XA=AY##.
    (Here ##X,\ Y, \ A## are square matrices of the same dimension.)

    Now in your case: apply this rule to ##B^{-1}AB=C## to get ##A=BCB^{-1}##.
     
  4. Mar 8, 2016 #3
    Ah. so essentially whatever order they are in the at the left is the same order they get applied to on the right?
     
  5. Mar 8, 2016 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    No: they do not have the same order on the two sides.Look again, carefully.
     
  6. Mar 9, 2016 #5
    Not understanding it completely. So say if I multiply the left side by some variable X and I put it to the left of whatever is already there, I have to do the same to the right? like say A=C, would be XA = XC?
     
  7. Mar 9, 2016 #6

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    Yes, this is correct. If A=C, then XA=XC.

    Your previous post was ambiguous.
     
  8. Mar 9, 2016 #7
    Ah. Thanks a lot!
     
  9. Mar 9, 2016 #8

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You are given that [itex]B^{-1}AB= C[/itex]. Knowing that matrix multiplication is not commutative, get rid of the "[itex]B^{-1}[/itex]" on the left by multiplying, by B, on both sides on the left: [itex]B(B^{1}AB)= BC[/itex]. Because matrix multiplication is "associative" that is the same as [itex](BB^{-1})AB= AB= BC[/itex]. And to get rid of the "B" or the right, multiply on both sides by [itex]B^{-1}[/itex] on the right to get [itex](AB)B^{-1}=A(BB^{-1})= A= BCB^{-1}[/itex]. It's just a matter of keeping track of which side you are on!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Matrix Equations
  1. Matrix equation (Replies: 4)

  2. Matrix equation (Replies: 7)

Loading...