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Name: CSU ID: Homework 2 September 4, 2015 Show work to receive credit. 1. (a) Given A defined below, find AT . (b) What are the dimensions of A and AT ? (c) How many pivots does A and AT have? (When finding the rref form of a matrix with more rows than columns, it may be necessary to fill in enough columns with 0 in each element so that the calculator will perform the operation.) A= −3 4 6 0 2 7 3 1 1 3 2 7 2. Defining basic matrix operations, find A−1 . That is, consider [A, I] → [I, A−1 ] 1 −1 −2 A = 2 −3 −5 −1 3 5 3. Show that if a square matrix A satisfies A2 − 5A + I = 0, then A−1 = 5I − A. 4. Simplify C −1 A−1 (BAT )T (B −1 )T C 5. Using the inversion algorithm discussed in class to find A−1 . You must identify E1 , . . . , E5 . 3 5 −6 4 A = −6 −9 21 31 −1 6. Find the inverse of each of the following matrices. For each matrix, how is the stucture of the inverse related to the structure of the original matrix? 2 −1 0 4 0 , A= 3 0 0 5 0 −3 7 4 2 , B= 0 3 0 0 6 0 −4 0 , C= 0 5 2 0 7 7. Suppose that ~u is a solution to A~x = ~b and that ~v is a solution to A~x = ~0. Show that ~u + α~v is a solution to A~x = ~b for any value of α.