State whether each of the following statements is True/False. Explain your answer. (Answers without any explanation will be awarded zero marks.) [max 50 words per statement]. Clearly label Part A & Part B in your answer. Part A [6 marks] Let Z be defined as: Z = Y—X(XTX)’1XT where XTX is non—singular. X & Y are both 3×3 matrices.
Statement I: 2 must be square. Statement ll: XTX must be square. Statement Ill: If the diagonal entries of both X and Y are 3 and X is a lower triangular matrix while Y is an upper triangular matrix,
then the determinant of XY is equal to 27. Part B [6 marks]
Consider the following function of two variables:
f(x,y) = 2y3 – Bxy — x2 By choosing specific values of x or y, we can construct new functions from f(x,y). For example, f(x,1) is the function constructed
from f(x,y) by specifying that y=1. So f(x,1) = 2(13) — 6x(1) — x2 = 2 — 6x — x2. Notice that f(x,1) is a function of only one variable, x. Statement I: The function f(O,y) has a saddle point.
Statement II: The function f(9,y) has a local minimum. Statement Ill: The function f(y,y) has two 2 critical points, of which one is an inflection point.
