Matrix Multiplication and Algebraic Properties of Matrix Operations

[hkus10]

1) If A = [aij] is an n x n matrix, the trace of A, Tr(A), is defined as the sum of all elements on the main diagonal of A, Tr(A) = the sum of (aii) from i=1 to n. Show each of the following:

a) Tr(cA) = cTr(A), where c is a real number

b) Tr(A+B) = Tr(A) + Tr(B)

c) Tr(A(Transpose)) = Tr(A)

 

2) If r and s are real numbers and A and B are matrices of the appropriate sizes, then proves the following:

A(rB) = r(AB) =(rA)B

 

I have been thinking these for a long time with no directions to approach. Please help!!!

[timo]

Oh come on. You've been working long and didn't figure out [math] {\rm Tr} (cA) = \sum_{i=1}^n ca_{ii} = c \sum_{i=1}^n a_{ii} = c \, {\rm Tr} (A) [/math] ? 1b, 1c work the same way. Question 2 very analogous.

[imatfaal]

Perhaps start by proving to yourself with simple 2x2 matrices; first with actual numbers and then with w,x,y,z. You should be able to do that without too much sweat, by simply working out both LHS and RHS and showing that they come to the same thing.