Linear Algebra Answers

write the matrix A[3,-1,1,-2]as a linear combination of A1=[1,1,0,-1],A2=[1,1,-1,0]and A3=[1,-1,0,0]

A boy was born on his father's 30th birthday. The sum of their ages is now 40 years. How old is the boy?

Prove that, if S is square, U is unitary, and U^(conjugate transpose)SU = T is upper triangular, then the eigenvalues of S and T are the same and S and T have the same trace. (Use the facts that det(AB) = det(A)det(B), and tr(ABC) = tr(CAB) = tr(BCA)).

Prove that the eigenvalues of an upper triangular matrix T are the entries of its main diagonal, so that the trace of T is the sum of its eigenvalues.

Shur's Lemma: For any square matrix S therer is a unitary matrix U such that
U^(conjugate transpose)SU = T is an upper triangular matrix.

Use Shur's Lemma to prove that
(a) If S^(conjugate transpose)=S then there is a unitary matrix U such that U^(conjugate transpose)SU is a real diagonal matrix.

(b) If S is real and and S^T = S then there is an orthogonal matrix O such that O^TSO is a real diagonal matrix.

Associated with each equation (Ax)_i=b_i in the system Ax=b there is a hyperplane H_i defined to be the subset of J-dimensional column vectors given by:

H_i = {x|(Ax)_i=b_i}.

Show that the ith column of A^(conjugate transpose) is normal to the hyperplane H_i; that is, it is orthogonal to every vector lying in H_i.

Ok so I have 3 matrix' and its A B C and I have to solve (-1 means inverse) for (A-1xB)-1xC so does (A inverse times B) the whole thing inverse times C = (B-1A)C?