1. Read the entire question through before beginning! The solution will need to be presented in stages in the different parts below. At a certain Community College, rst-year students in a particular program must take a 100 level math class, a 100 level English class and a philosophy class. In order to satisfy this requirement in any given quarter, the college needs to make 8 more sections of math classes available than sections of English classes, but only half as many sections of philosophy classes as sections of English classes. While planning for Fall quarter, it became clear that several factors (classroom avail- ability, time of day, instructor availability) would limit the total number of sections of these three courses to 63. How many sections of each type of class should the college make available during fall quarter? (a) Write the problem as a system of three linear equations and three variables. (1)) Write the system of linear equations you wrote in part (a) as an augmented matrix. (c) Use elementary row operations to nd the reduced echelon form of the matrix you wrote in part (b). (d) Interpret the reduced echelon form of the matrix you found in part (c) as the answer to the original question and write out the result as a short sentence. if? 01 2-1 $333 2-A= B=31 0:00 D: 101 21 15 401 200 210 Evaluate the following matrix expressions. If an expression cannot be evaluated, give a detailed reason. (a) 2A8 3D (b) D(C + B) (c) Find A"1 3. Solve the matrix equation Ax = b below by using an inverse matrix. Be sure to show the steps you took to find the inverse matrix. Briefly describe what your answer means or what it represents. 0 C1 0 1 34. Solve the matrix equation Ax = b below by the mechanical (not theoretical) technique of your choice. Notice that this is the same matrix equation as. in question 2 with the exception of the vector b. Be sure to show the steps you take during any matrix multiplication. On this problem, you may use a calculator to do the matrix manipulation, but if you do, you must explain what you did on the calculator and why. Bonus: How could you have used the invertible matrix theorem to reach the same con- clusion? Be sure to include statements of any restrictions and how you know they are met. 0 3 2 1'1 0 1 4 2 3:2 = 0 3 4 1 $3 0