Determinant proof

  • Thread starter KataKoniK
  • Start date
  • #1
168
0

Main Question or Discussion Point

Is this proof correct?

Show that det(A + B^T) = det(A^T + B)

det(A + B^T) = det(A^T + B)
det(A) + det(B^T) = det(A^T) + det(B)
det(A) + det(B) = det(A) + det(B)

Thanks.
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,833
955
det(A+B) is not, in general, equal to det(A)+ det(B).

What is true is that (A+B)T= AT+ BT so that, in particular (AT+ B)T= A+ BT.
 
Last edited by a moderator:
  • #3
AKG
Science Advisor
Homework Helper
2,565
4
Well your "proof" is backwards for one, and second it's invalid. It seems you're suggesting that det(X + Y) = det(X) + det(Y). What if X = I2 (the 2x2 identity matrix) and Y = -I2?

The determinant is not a linear function, however transposition is linear. Use that fact plus the trivial fact that matrix addition is commutative, plus the fact that the det(AT) = det(A), to get the desired result.
 
  • #4
168
0
Hmm, so is this the correct method of doing it?

det(A + B^T) = det(A^T + B)
det(A + B^T)^T = det(A^T + B)^T
det(A^T + B) = det(A + B^T)

I can't do this, correct? Thanks for the hints though.
 
  • #5
TD
Homework Helper
1,022
0
Well, I think it will be enough if you do it on one side :wink:

det(A + B^T) = det(A^T + B)
det(A + B^T)^T = det(A^T + B)
det(A^T + B) = det(A^T + B)
 
  • #6
AKG
Science Advisor
Homework Helper
2,565
4
det(AT + B) = det((AT + B)T) = det(ATT + BT) = det(A + BT)
 
  • #7
HallsofIvy
Science Advisor
Homework Helper
41,833
955
This proof, as TD wrote:
det(A + B^T) = det(A^T + B)
det(A + B^T)^T = det(A^T + B)
det(A^T + B) = det(A^T + B)
which starts with what you want to show and arrives at a true statement is valid if you make sure each step is reversible! (That's often called "synthetic proof".)

But AKG's suggestion:
det(AT + B) = det((AT + B)T) = det(ATT + BT) = det(A + BT)
is much nicer!
 
  • #8
168
0
Thanks everyone!
 

Related Threads on Determinant proof

  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
2
Views
17K
Replies
6
Views
11K
  • Last Post
Replies
2
Views
2K
Replies
8
Views
40K
Replies
4
Views
2K
Top