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!

 

[Mark44]

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!

For 1, you show the definition of Tr(A). What is Tr(cA)? Tr(A + B)? Tr(A^T)?
For 2, look at a typical element of the matrices A(rB), r(AB), and (rA)B.

 

[HallsofIvy]

You don't do math problems by sitting and staring at a piece of paper so if you have "been thinking about these for a long time", you must have tried something. Show us what you have tried.