hkus10

1) show that if AB = AC and A is nonsingular, then B = C. 2) show that if A is nonsingular and AB = 0 for an n x n matrix B, then B = 0. 3) Consider the homogenous system Ax=0, where A is n x n. If A is nonsingular, show that the only solution is the trivial one, x=0. 4) Prove that if A is symmetric and nonsingular, then A^-1 is symmetric. Please help and show all your work or at least give me some directions! Thanks

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!!!

Let C1 and C2 be circles in the plane. Describe the number of possible points of intersection of C1 and C2. It is 4?

1) Is the matrix [upper row 3 0 and lower row 0 2] a linear combination of the matrices [upper row 1 0 and lower row 0 1] and [upper row 1 0 and lower row 0 0]? Justify your answer. Is it I just have to add the two matrices to see if they are equal the matrix, [upper row 3 0 and lower row 0 2]? 2) Show that the linear system obtained by adding a multiple of an equation in (2) to another equation is equivalent to (2). How to show that? Thanks!

solve each given linear system by the method of elimination 2x-3y+4z=-12 x-2y+z=-5 3x+y+2z=1 How to solve this problem. What should I do first?