Suppose m linear systems Ax*'1 ' b 1 ' 1 ', P 1,2,... , m, are to be solved, each with the n x n coefficient matrix A. a. Show that Gaussian elimination with backward substitution applied to the augmented matrix [ A : b (1)b<2) b"'" ] requires I , 7 1 -n + inn -n multiplications/divisions and 1 , 7 1 9 1 -n + mn n mn Hn additions/subtraction b. Show that the Gauss-Jordan method (see Exercise 14, Section 6.1) applied to the augmented matrix A : b (1)b <2) b"'" ] requires and 1 , ,1 -zr + mn n multiplications/divisions 2 2 I , , /1 -n + (m \)n +(-?) additions/subtractions. c. For the special case b"" = 0 1 pth row, for each p = ,m, with m = n, the solution x 1 ''' is the pth column of A Show that Gaussian elimination with backward substitution requires 4 1 -n3 -n multiplications/divisions and 4 , 3 , 1 -n n Hn additions/subtractions 3 2 6 for this application and that the Gauss-Jordan method requires 3 , I -n n multiplications/divisions 2 2 F and 3 , 2 1 -n 2n Hn additions/subtractions. 2 2 d. Construct an algorithm using Gaussian elimination to find A -1 but do not perform multiplications when one of the multipliers is known to be 1 and do not perform additions/subtractions when one ofthe elements involved is known to be 0. Show that the required computations are reduced to n 3 multiplications/divisions and n 3 2n2 + n additions/subtractions. e. Show that solving the linear system Ax b, when A -1 is known still requires n 2 multiplications/divisions and n 2 n additions/subtractions. f. Show that solving m linear systems Ax1 ''' = b (p) , for /> = 1,2,... , m, by the method x (/' ) = A-'b'''1 requires mn2 multiplications and m(n2 n) additions if A -1 is known. g. Let A be an n x n matrix. Compare the number ofoperations required to solve n linear systems involving A by Gaussian elimination with backward substitution and by first inverting A and then multiplying Ax = bby A -1 , forn = 3, 10, 50, and 100. Is it ever advantageous to compute A -1 for the purpose of solving linear systems?

Imperfect is used when… -referring to actions repeatedly or happened over period of time Phrases that can signify a repeated action -“cada” or “todas” (every) day, year, etc. -“nunca” never -“siempre” always -“a menudo” often, a veces sometimes -“por un rato” or for a while -“generalmente” (generally) or “frecuentamente” (frequently) -Actions that “set the stage” in a narrative “Hablaba a Juan cuando se cayó” “I was talking to Juan when I fell” -Referring to a physical or mental state of well-being Ex. “Juana estaba enfermo” Meanwhile, regular past tense is used for one time actions or actions that have been completed