Question:

2 1 3 A = ll] 7 1D . 4 5 {a} Find the FLU factorization of A. Justify all your steps. Hint. You may {and should} use, without proof, the properties of lGaussian transformation matrices presented in Problem T of Module 2 practice problem set [click here}. Be sure to give a clear reference to the properties used in your solution. {b} {:hmp'llte det[P}, where P is the permutation matrix found in part {a}. [e] l[C'AZImpnte deth} using Me is! and 1b! alone. The result obtained by a direct calculation will not be aecepted. Be sure to justify all your steps clearly. 7. (Gaussian transformation matrices; Su20 final exam) , Let {e; e R" | j e N[1, n] } be the standard unit basis of R", Le., el = (1, 0, 0, . .. , 0) , e2 = (0, 1, 0, ... , 0) , ..., en = (0, 0, 0, . .. , 1) . In this problem, we denote by G, the Gaussian transformation matrix of the form G; = It _ ajeeJ. i=j+1 In addition, let P(i, j) e Rox be the elementary permutation matrix obtained by interchang- ing the i-th and the j-th rows of the same-sized identity matrix. (a) Let 1 j aijere] i=j+1 (c) Let j axe;el. i=j+1 i=k+1 2 (d) Use the previous parts to find PLU factorization, PA = LU, by hand. A = 10