Unformatted text preview: = 0. F. Let L = { A ∈ M n ( R )  all row sums of A are 1, that is A i 1 + A i 2 + ⋅ ⋅ ⋅ + A in = 1 for all i }. Show that A ∈ L ⇔ [1, 1, . . . , 1] T ∈ R n is a solution of AX = X . Use this to show: A , B ∈ L ⇒ AB ∈ L . G. Show that A ∈ M n ( R ) has rank n ⇔ the row reduced echelon form of A is I n . H. If A ∈ M n ( R ) has rank n then the systems AX = B is consistent for any B ∈ R n (Use G. ) I. If A , B ∈ R m x n have the same row reduced echelon form E , then show that the homogeneous systems AX = 0 n x1 and BX = n x1 are equivalent. J. Describe those a b c " # $ $ % & ' ' ∈ R 3 so that the system 1 1 –1 2 1 1 7 2 " # $ $ % & ' ' X = a b c " # $ $ % & ' ' is consistent. Hand In 2A) p. 139 #8; 10; 11; B; pp. 149 –150 #18; 23; 24; F 2B) C; G; H; J Due Wednesday September 7...
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 '07
 Guralnick
 Differential Equations, Linear Algebra, Algebra, Real Numbers, Equations, #, tc, 1 1 0%, $ 1 7 2

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