# Matrix Multiplication and Algebraic Properties of Matrix Operations

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

Mark44
Mentor
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