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Assign 1 - MATH 225 ASSIGNMENT 1 Work the following...

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MATH 225 ASSIGNMENT 1 Work the following problems. These are not to be handed in. In the problems below, R denotes a ring, Z denotes the set of integers, and R denotes the set of real numbers. §2.1 p. 117 #1 – 7; pp. 118 – 119 #1 – 15, 26, 27. A. Let A M n ( Z ) have all its entries zero except for one 1 each row, one 1 in each column, and at most one 1 on the diagonal. Give examples of such matrices for n = 2, 3, 4, 5. In general, show that for any matrix A with this property, A T also satisfies this property. B. A general symmetric 2 × 2 matrix has the form: a b b a " # $ % & ' . Write out a general symmetric 4 × 4 matrix and general skew-symmetric 4 × 4 matrix. C. Show that for any matrix A , ( A T ) T = A . If B R 7x9 must B = A T for some matrix A ? Why, in which case describe A , or why not? D. For any given A M 2 ( R ) let B M 2 ( R ) be defined by B ij = A ij 2 . If tr( A ) = tr( B ) = 0 show that A 11 = A 22 = 0. Does this remain true if instead B ij = A ij 3 ? E. For each of the definitions below of A M n ( R ), determine tr( A ) and whether A is symmetric, skew-symmetric, or neither: i)
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