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.
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- 6). Compute the directional derivative of function f(x,y) = xy³ + 2.x²y at the point (2, 1) in the direction (,, ).1 1 a) Let A = 1 -1 -1 1 (i) Compute the value of det (A). (ii) For what values of x will the matrix be singular? b) Write down the quadratic form that corresponds to the following matrix. 2 4 2 -1 -3 .4 -3 1Solve the following problems by using the inverse of the matrix involved. (a) An automobile factory producess two models, A and B. Model A requires 1 labor hour to paint and 1 labor hour to polish; model B requires 1 labor hour for each 2 process. During each hour that the assembly line is operating, there are 80 labor hours available for painting and 65 labor hours for polishing. How many of each model can be produced each hour if all the labor hours available are to be utilized? (b) Suppose each model A requires 14 widgets and 10 shims and each model B requires 10 widgets and 7 shims. The factory can obtain 1050 widgets and 745 shims each hour. How many cars of each model can it produce while using all the parts available? (a) Model A: Model B: (b) Model A: Model B:
- f(x1,x2) 4 2 0 -2 -4 −1 2 0 1 1 0 2 -1 X1 3-2 x2 Figure 2: Graph of the function f(x1, x2) = 2x1 = x²-x+3 Consider the following function: f(x1, x2) = 2x1 - x-x+3. - For your visual convenience, a portion of the function is depicted in Figure 2. (a) Compute the gradient vector of f. (b) Compute the Hessian matrix of f. (c) Determine all points that yield stationary values of f. (d) Use the second-order conditions to determine whether each of the stationary val- ues of f is a maximum, a minimum, or neither.For what value(s) of x is the function f(x) NOT continuous? A x = 0 and x = 3 C. x = -3 only Select one: = 4 2x² — 6x B. x = 3 only D. x = 3 and x = 4Traders often buy foreign currency in the hope of making money when the currency’s value changes. For example, on April 15, 2015, one U.S. dollar could purchase 0.9428 euro, and one euro could purchase 126.457 yen. Let f(x) represent the number of euros you can buy with x dollars, andlet g(x) represent the number of yen you can buy with x euros.(a) Find a function that relates dollars to euros.(b) Find a function that relates euros to yen.(c) Use the results of parts (a) and (b) to find a function that relates dollars to yen. That is, find (g ∘ f) (x) = g(f(x) ).(d) What is g(f(1000) )?
- For the matrix A = 4 3 " 6 what is the element in the third row 3 2 and third column of the Matrix of Cofactors A. 5 C. 4 B. 4 D. -7 Select one: OD O C OB OAWrite the following system of equations in the matrix form AX = B;where. Quesiton is in the imageRefresh your Math & Graphing Skills SOP-A D 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. X-axis 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 The average grade received" and Y-axis should be labeled "The number of hours spent studying". Scenario As the number of hours spent studying rises, the average grade received rises. As the number of hours spent studying falis, the likelihood of getting an A falls. As the number of hours spent watching TV rises, the average grade received falls. True True or False: Line B has a slope of infinity. O False www Relationship Line Y
- Compute the first and second derivatives of the following functions of x. In each case, determine whether the function has any local extremes (maxima or minima) on the domain of nonnegative real numbers – i.e. for x E [0, 0). a. f(x) = 4 + 3x? b. f(x) = x (4x – 3) (9 - 5x) с. f(x) — x2 d. f(x) = In(x) -. what is a Function? Discuss the following concepts of functions with the help of an economic example : 4) Exponential function 5) Logarithmic Function5. Suppose a quadratic function is given by y = f(x) = x29x + 8. (a) Does this function represent an upward or downward facing parabola? Explain. Find the x - coordinate of the vertex (or the equation of the vertical line that intersects the parabola at the vertex) of the parabola.(b) Find the roots of the quadratic function. 1 (c) Using information from (a) and (b) draw a well labelled graph of the parabola (in particular, label the vertex and the roots).(d) In the same graph as (c), draw the parabola if it was instead represented by the function f(x) = x2-9x + 21. How many roots does this parabola have?6. Suppose an inverse demand function is given by p =10 - qd and an inverse supply function is given by p = 4+ 2qs. (a) What is the equilibrium price and quantity combination? (b) What happens when the price is above or below the equilibrium price? 7. How many solutions does a system of two equations have? Explain clearly and succinctly.