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

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The discussion revolves around proving properties of the trace of matrices, specifically that Tr(A + B) = Tr(A) + Tr(B) and Tr(AB) = Tr(BA). Participants emphasize using the definition of trace in sigma notation to demonstrate these properties. The problem also includes a third part related to a 2x2 matrix, asserting that A^2 - Tr(A)A + det(A)*I2 = O. The original poster expresses a lack of confidence in solving these problems and requests detailed solutions. The conversation highlights the importance of understanding matrix operations and their implications on the trace function.
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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")
 
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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...
 
Prove It said:
a) should be easy...

Okay, but can you help me with the rest?
 
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 $\text{tr}(AB)$?
 

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