Matrices Show that Tr(A + B) = Tr(A) + Tr(B).

  • MHB
  • Thread starter MaXiiMo
  • Start date
  • Tags
    Matrices
In summary, the conversation is discussing a problem related to the trace of an n x n matrix A and asking for help in solving it. The problem involves showing that Tr(A + B) = Tr(A) + Tr(B), Tr(AB) = Tr(BA), and A^2 - Tr(A)A + det(A)*I2 = O for a 2 x 2 matrix A. The respondent suggests using the definition of trace in sigma notation to solve the problem.
  • #1
MaXiiMo
4
0
I do not have any work to show as I am not skilled enough to solve this problem as of yet. I really do need an answer to the question though. I know this is a long shot but I am desperate at the moment, so please do provide the solution with steps to the problem below. Many thanks.

Problem) The trace of an n x n matrix A is:

Tr(a) = a11 + a22 + ... + ann.(a) Show that Tr(A + B) = Tr(A) + Tr(B).

(b) Show that Tr(AB) = Tr(BA).

(c) Show: For a 2 x 2 matrix A, we have

A^2 - Tr(A)A + det(A)*I2 = O.

(I believe I2 is representative of "Identity matrix 2")
 
Mathematics news on Phys.org
  • #2
MQ1993 said:
I do not have any work to show as I am not skilled enough to solve this problem as of yet. I really do need an answer to the question though. I know this is a long shot but I am desperate at the moment, so please do provide the solution with steps to the problem below. Many thanks.

Problem) The trace of an n x n matrix A is:

Tr(a) = a11 + a22 + ... + ann.(a) Show that Tr(A + B) = Tr(A) + Tr(B).

(b) Show that Tr(AB) = Tr(BA).

(c) Show: For a 2 x 2 matrix A, we have

A^2 - Tr(A)A + det(A)*I2 = O.

(I believe I2 is representative of "Identity matrix 2")

a) should be easy...
 
  • #3
Prove It said:
a) should be easy...

Okay, but can you help me with the rest?
 
  • #4
Hi MQ1993, :)

Both (a) and (b) can be shown by using the definition of trace in sigma notation. If $ \displaystyle \text{tr}(A)=\sum_{i=1}^{n}a_{ii}$, what is the definition of $\text{tr}(AB)$?
 

FAQ: Matrices Show that Tr(A + B) = Tr(A) + Tr(B).

What are matrices and how are they used in mathematics?

Matrices are rectangular arrays of numbers or variables, commonly used in mathematical calculations and in representing linear transformations. They are often used to solve systems of linear equations and to represent geometric transformations.

What is the trace of a matrix?

The trace of a matrix is the sum of its diagonal elements. It is often denoted by "Tr" and is a measure of the sum of all the elements along the main diagonal of a square matrix.

How do we show that Tr(A + B) = Tr(A) + Tr(B) for two matrices A and B?

To show that Tr(A + B) = Tr(A) + Tr(B), we can use the properties of matrix addition and the fact that the trace of a matrix is the sum of its diagonal elements. By adding the corresponding elements of A and B, we can see that the resulting matrix will have the same diagonal elements as A and B, resulting in the same trace.

Can we use this property for any size of matrices?

Yes, this property holds true for any size of matrices as long as they are square matrices. This means that they have the same number of rows and columns. The trace of a non-square matrix is not defined, so this property only applies to square matrices.

Why is the property Tr(A + B) = Tr(A) + Tr(B) important in linear algebra?

This property is important because it allows us to simplify calculations involving matrices. Instead of calculating the trace of a sum of matrices, we can simply add the traces of each individual matrix. This property also helps in proving other important theorems and properties in linear algebra.

Similar threads

MHB Is Every Statement Correct in Defining an Onto Function?
Replies
6
Views
1K
Multiplication of Operators in Quantum Mechanics
Replies
3
Views
993
MHB Estimate the cost for tyres using the mean
Replies
9
Views
2K
MHB How can we approach a seemingly senseless binomial expansion problem?
Replies
4
Views
2K
I ##Tr([Q,P])## and Ballentine Problem 6.3
Replies
31
Views
2K
Optimization: Formulation of the dual of a semi-definite program (SDP)
Replies
1
Views
767
Solving Computing a Trace Properties: Any Help Appreciated
Replies
1
Views
1K
A Proving BF action is a difference of Chern-Simons actions
Replies
0
Views
610
A Question about a property of a matrix of transition probabilities
Replies
6
Views
1K
Prove trace of matrix: Tr(AB) = Tr(BA)
Replies
5
Views
10K

Hot Threads

Recent Insights

Back
Top