Let A and B be n X n orthogonal matrices. (a) Prove that A{AT + BT)B = A + B. (b) Use part (a) to prove that, if det A + det B = 0, then A + B is not invertible.

Consider matrix A = Answer: 0 1 (₁2) G -1 2 and vectors x = (¹2₂) and y = (1). . Calculate matrix B = (A – AT)² and vector w = Bx. Then, evaluate the scalar product w · y. [Please enter your answer numerically. You will be marked correct as long as what you enter is within 0.5 of the correct answer.]

Let A be an m x n matrix, v be a vector in R", and c be a scalar. Show that A(cv) = c(Av).

1 Let A = 4 1 a) Write A as a product of elementary matrices. b) Using A-1, solve XA= B for the matrix X where B =(-1 2).