Linear algebra question

  • Thread starter kwal0203
  • Start date
  • #1
69
0

Homework Statement



Assuming that all matrices are [itex]n\times n[/itex] and invertible, solve for [itex]D[/itex].

[itex]C^{T}B^{-1}A^{2}BAC^{-1}DA^{-2}B^{T}C^{-2}=C^{T}[/itex]

The Attempt at a Solution



I tried to group all like terms and simplify. I'm pretty sure this is not allowed but I'm not really sure how to approach this question. Thanks a lot any help appreciated!

[itex]C^{T}C^{-1}C^{-1}C^{-1}B^{-1}BB^{T}AAAA^{-1}A^{-1}D=C^{T}[/itex]

[itex]C^{T}C^{-1}C^{-1}C^{-1}IB^{T}IIAD=C^{T}[/itex]

[itex]((C^T)^{-1}C^{T})C^{-1}C^{-1}C^{-1}B^{T}AD=C^{T}(C^T)^{-1}[/itex]

[itex]I(C^{-1}C^{-1}C^{-1}CCC)(B^{T}(B^T)^{-1})(AA^{-1})D=ICCC(B^T)^{-1}A^{-1}[/itex]

[itex]D=C^{3}(B^T)^{-1}A^{-1}[/itex]

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
36,165
6,783
No, you can't change the order like that. What you can do is multiply both sides by the same thing at the same end. I.e. You can go from Y = Z to XY = XZ or to YX = ZX. Start with multiplying both sides on the left by C-T. Use the fact that you can change the order when the two matrices being multiplied are inverses of each other: X-1X = I = XX-1
 
  • #3
Dick
Science Advisor
Homework Helper
26,260
619

Homework Statement



Assuming that all matrices are [itex]n\times n[/itex] and invertible, solve for [itex]D[/itex].

[itex]C^{T}B^{-1}A^{2}BAC^{-1}DA^{-2}B^{T}C^{-2}=C^{T}[/itex]

The Attempt at a Solution



I tried to group all like terms and simplify. I'm pretty sure this is not allowed but I'm not really sure how to approach this question. Thanks a lot any help appreciated!

[itex]C^{T}C^{-1}C^{-1}C^{-1}B^{-1}BB^{T}AAAA^{-1}A^{-1}D=C^{T}[/itex]

[itex]C^{T}C^{-1}C^{-1}C^{-1}IB^{T}IIAD=C^{T}[/itex]

[itex]((C^T)^{-1}C^{T})C^{-1}C^{-1}C^{-1}B^{T}AD=C^{T}(C^T)^{-1}[/itex]

[itex]I(C^{-1}C^{-1}C^{-1}CCC)(B^{T}(B^T)^{-1})(AA^{-1})D=ICCC(B^T)^{-1}A^{-1}[/itex]

[itex]D=C^{3}(B^T)^{-1}A^{-1}[/itex]

Homework Statement





Homework Equations





The Attempt at a Solution


Don't assume the matrices commute. You can't interchange the order like you did. Aside from the fact this problem is needlessly complex, just use patience and cancel each matrix M by M^(-1) on the appropriate side.
 
  • #4
69
0
Thanks guys, but what happens when I get down to D after I multiply each term preceding it by its inverse?
 
  • #5
69
0
Ahhhh, multiply on the right!
 
  • #6
Dick
Science Advisor
Homework Helper
26,260
619
Ahhhh, multiply on the right!

Right! No pun intended.
 
  • #7
69
0
Right! No pun intended.


Lol thanks
 

Related Threads on Linear algebra question

  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
1
Views
745
  • Last Post
Replies
4
Views
874
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
806
  • Last Post
Replies
1
Views
845
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
2
Views
716
Top