# HW 1 - E ij ij = 1(the i-j entry is 1 and all other entries...

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MATH 225 HOMEWORK 1 pp.130 – 133 True-False 1 – 8; # 1 – 4, 8, 10, 12, 13, 15 – 17, 19, 26, 27, 29, 30 ( R = R ), 32 A. If A = 1 1 -2 1 0 1 1 0 0 " # \$ \$ % ' ' compute A 3 A 2 + A . B. If A = a b d 0 c e 0 0 f " # \$ \$ % ' ' show that ( A aI 3 )( A cI 3 )( A fI 3 ) = 0 3x3 . C. Let B n R n x n be the matrix all of whose entries are 1. For any C R n x n describe B n CB n . D. Describe all A = a b c d ! " # \$ R 2x2 so that A 2 = A . Recall that for A R n x n , the trace of A , tr ( A ), is the sum of the diagonal entries of A . E. For A , B R n x n and s R , show that tr ( A + B ) = tr ( A ) + tr ( B ) and tr ( sA ) = s tr ( A ). F. If A R n x n then show: i) AA T is symmetric, and ii) tr ( AA T ) = 0 A = 0 n x n . G. Let A = diag(1, 2, 3, . . . , n ) M n ( R ). If B M n ( R ) so that AB = BA , show that B is a diagonal matrix. Show that this result is not true for A = diag(1, 1, 3, 4, . . . , n ). H. Let A , B R 4x4 so that row 1 ( B ) = 2row 2 ( A ) + 3row 2 ( A ) – row 4 ( A ), row 2 ( B ) = row 1 ( A ) + 3row 3 ( A ), Row 3 ( B ) = row 1 ( A ), and row 4 ( B ) = row 1 ( A ) + row 2 ( A ) + row 3 ( A ) + row 4 ( A ). Find P R 4x4 so that PA = B . I. Let E ij R n x n be the matrix with (
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Unformatted text preview: E ij ) ij = 1 (the i-j entry is 1), and all other entries are 0. Show that E ij E st = E it if j = s and that E ij E st = 0 n × n when j ≠ s . Show that every A ∈ R n x n is a linear combination of the various E ij . J. Show that A ∈ M n ( R ) satisfies tr ( A ) = 0 if and only if A is a linear combination of: { E ij | i ≠ j } ∪ { E jj – E nn | 1 ≤ j ≤ n – 1}. K. Given 1 ≤ i < j ≤ n , describe P ∈ M n ( R ) so that for any A ∈ R n x m , B = PA is the matrix A with rows i and j interchanged: row i ( B ) = row j ( A ), row j ( B ) = row i ( A ), and row k ( B ) = row k ( A ) for all k ≠ i , j . Show P has the same effect on the columns of any C ∈ R pxn . Hand In 1A) p.131. #10; 16; 26; 30; B; C; D; Fii) 1B) G; H; J; K Due Wednesday, August 31...
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