Name: CSU ID: Homework 1 January 27, 2016 1. Referring to S1.1, ]10, find a general solution to the system. 2. Define the matrix E1 and E2 that bring the following system to upper triangular form. Using backward substituion, solve the system. −17 −2 5 4 x = −54 −6 18 16 ~ 20 2 10 22 3. S1.1 ]16(b) 4. S1.1 ]26 5. S1.1 ]TF 6. Suppose that A is a 5 × 6 matrix that is blocked off as given below. If A11 is 2 × 3, what are the dimension of A12 , A21 , A22 ? " A= A11 A12 A21 A22 # 7. S1.2 ]6 8. S1.3 For ]12, define the matrix equation and solve. 9. S1.3 ]14 10. Given the system below, complete the following. (a) Augment the coefficient matrix with the right hand side vector and find the RREF(NOT by hand). (b) Define the solution to the system in vector form as discussed in class. (c) How many “free variable” are there? −2 4 4 −1 2 0 −6 18 16 −3 9 3 2 10 22 3 13 17 −4 −2 −2 0 −7 −6 x= ~ −9 −15 93 −40 " # " # 4 1 a b 11. Let A = . Define matrices of the form M = such 2 −1 c d that AM = M A. Hint: There are 4 unknowns. AM = M A can be rewritten as a matrix system with elements of M appearing in a vector of unknowns (take M and have the vector be 4 × 1). 12. Consider the matrix A 2 7 −1 4 6 −7 A = −4 −17 8 19 9 24 A can be brought to RREF by operators Ej , j = 1 : 5, where E1 eliminates the (2,1) and (3,1) positions, E2 eliminates the (3, 2) position, E3 replaces the diagonal elements with 1, E4 eliminates the (2, 3) and (1, 3) positions, and E5 eliminates the (1, 2) position. Determine the Ej . The answers should be defined with rational numbers.

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# Name: Homework 1 CSU ID: January 27, 2016