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HW 2(1) - = 0 F Let L = A ∈ M n R | all row sums of A are...

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MATH 225 HOMEWORK 2 p. 138 True/False #1 – 6; pp. 138 – 139 #1, 4, 6 – 11 p. 149 True/False # 4 – 9; pp. 149 – 150 #16 – 25 (Find the reduced row echelon form and rank.) Below, R is the field of real numbers. A. Let A , S , T M n ( R ) and B R n . If ST = I n show that if C R n is a solution of AX = B then TC is a solution of ( AS ) X = B . B. If A , S M n ( R ) with SA = I n = AS show that for any B R n the system AX = B is consistent and has a unique solution. C. If for some A M n ( R ) and any B R n , AX = B has a unique solution, show that AS = I n for some S M n ( R ). (Look at the solutions of AX = col j ( I n ) and think about matrix multiplication.) D. Given A M n ( R ) and r R , show that the solutions in R n of AX = rX are the solutions of some homogeneous system of linear equations (note that X = I n X ). E. Let S = { α 1 , . . . , α k R 1x n }. Describe a homogeneous system of linear equations whose solutions are exactly those β R n satisfying α 1 β = 0, α 2 β = 0, . . . ,
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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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