7. Consider an n x n matrix A that is not a null matrix. Suppose that A is idempotent, and define matrix B by B = I −k²A, where k € R is a parameter. Which of the options below is the set of ALL values k such that B is idempotent? a) R b) {0} {0,1} d) {-1,0} e) {-1,0, 1} f) {-3,0,3} g) The set of all positive whole numbers. h) None of the above.

For each of the following scenarios, indicate whether the relationship between the two variables is positive or negative, as well as which line on the previous graph has a slope that reflects this type of relationship.   Hint: The X-axis and Y-axis on the graph are not labeled intentionally. You need to substitute the variables from each scenario for the horizontal and vertical axis. For example, in the first scenario, X-axis should be labeled “ ice-cream" and Y-axis should be labeled "The temperature". Relationship options: positive, negative Line options: A, B, C, D   True or False: Line D has a slope of infinity.

Consider the linear spline function f(x) = 1 + x + h1(x, 2) where h1(x,2) = x-2 if x>2 and 0 otherwise. What is the slope of f at the point x=-1?

2. (This problem derives the general result that any polynomial function is of a lower asymptotic order than any exponential function.) First, we prove a special case. Suppose that we have two functions, f(n) = n² and g(n) = 2". We will prove that f(n) = O(g(n)). • port (a). (c) Show that, for any n ≥ no, it must be the case that f(n) ≤ cog(n) = f(n+1) ≤ cog(n+1). Precisely state the argument by induction that n² = 0(2¹). Now, define f(n) = nk where k ≥ 1, g(n) = a", where a ≥ 1. (d) (Slightly harder). Argue that, for any k and a, there exists some n₁ such that : (1₁+¹)* < a.