Matrices

MQ1993

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Feb 24, 2015
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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")

 
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")

This is just a matter of working through the problem with the definition. For example, take (a) and let C equal A+B, then
cij = aij + bij
so Tr(C) = sum(cii) = sum(aii + bii) = ? = Tr(A) + Tr(B)
now work through the other problems.
(b) If C = AB what is cii. If D = BA, what is dii
(c) Note that A2 - Tr(A) A = A [A - Tr(A)] and work it and the rest out.

If you can't do these then review the rules of matrices such as here
http://www.freemathhelp.com/matrix-multiplication.html
or do a web search such as
https://search.yahoo.com/yhs/search?p=matrix+multipication&ei=UTF-8&hspart=mozilla&hsimp=yhs-001
 
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