Symmetric Matrix Transpose: ABC^T ≠ CBA?

In summary, the conversation discusses the question of whether (ABC)^T is equal to CBA when A, B, and C are all symmetric matrices. It is mentioned that according to the solutions manual of Gilbert Strang Linear Algebra, the answer is no. However, the person in the conversation doubts this answer and suggests posting the full question and answer from the book. The final conclusion is that ABC is not equal to CBA and no one is claiming that they are.
  • #1
Ara macao
27
0
[tex](ABC)^T, A,B,C[/tex] are all symmetric, then why isn't [tex](ABC)^T = CBA[/tex]? If you consider that [tex](ABC)^T = (C^T)(B^T)(A^T)[/tex] and in symmetrix cases, then [tex]C^T = C[/tex] and so on...?

(Latex edit by HallsofIvy)
 
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  • #2
Who says that (ABC)^T is not CBA when all three are symmetric?
 
  • #3
The solutions manual to Gilbert Strang Linear Algebra...
 
  • #4
Right, why don't you post the full question and the full answer from this book? I mean, is the question:

Q. if A,B, and C are symmetric does (ABC)^T = CBA?
A. No.
 
  • #5
Right, why don't you post the full question and the full answer from this book? I mean, is the question:

Q. if A,B, and C are symmetric does (ABC)^T = CBA?
A. No.
 
  • #6
Yes, that is the case
 
  • #7
Then the asnwer book is wrong, if that is the precise statement of the question.
 
  • #8
ABC [tex]\neq[/tex]CBA
 
Last edited:
  • #9
And no one is claiming that they are equal.
 

1. Why is ABC^T not equal to CBA?

The reason for this is because the order in which matrices are multiplied matters. In the first case, matrix A is multiplied by matrix B, and the transpose of matrix C is multiplied on the right. In the second case, the transpose of matrix B is multiplied on the left, and matrix A is multiplied by matrix C. This changes the overall result of the matrix multiplication, making ABC^T not equal to CBA.

2. Can you provide an example to illustrate this?

Yes, let's take the following matrices as an example:

A = [1 2; 3 4]
B = [5 6; 7 8]
C = [9 10; 11 12]

Using the first order of multiplication, ABC^T would result in:

ABC^T = [37 54; 81 118]

While using the second order of multiplication, CBA would result in:

CBA = [39 54; 45 66]

As we can see, the two results are not equal.

3. Is there a specific condition where ABC^T can be equal to CBA?

Yes, for the two matrices to be equal, A, B, and C must be symmetric matrices. This means that they are equal to their own transpose, and the order of multiplication does not matter. In this case, ABC^T would be equal to CBA.

4. How is a symmetric matrix defined?

A symmetric matrix is a square matrix that is equal to its own transpose. This means that for a matrix A, A^T = A. In other words, the elements of the matrix are symmetric along the main diagonal.

5. What are the applications of symmetric matrix transpose?

The symmetric matrix transpose has various applications in mathematics, engineering, and computer science. Some examples include solving systems of linear equations, calculating eigenvectors and eigenvalues, and performing operations in computer graphics and machine learning algorithms.

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