# HW2_sol - 14 CHAPTER 1 Conversely suppose there exists a...

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14 CHAPTER 1 Conversely suppose there exists a nonsingular matrix M such that B = MA . Since M is nonsingular it is row equivalent to I . Thus there exist elementary matrices E 1 ,E 2 ,...,E k such that M = E k E k - 1 ··· E 1 I It follows that B = = E k E k - 1 E 1 A Therefore B is row equivalent to A . 27. (a) The system V c = y is given by 1 x 1 x 2 1 x n 1 1 x 2 x 2 2 x n 2 . . . 1 x n +1 x 2 n +1 x n n +1 c 1 c 2 . . . c n +1 = y 1 y 2 . . . y n +1 Comparing the i th row of each side, we have c 1 + c 2 x i + + c n +1 x n i = y i Thus p ( x i )= y i i =1 , 2 ,...,n +1 (b) If x 1 ,x 2 ,...,x n +1 are distinct and V c = 0 , then we can apply part (a) with y = 0 . Thus if p ( x c 1 + c 2 x + + c n +1 x n , then p ( x i )=0 i , 2 The polynomial p ( x ) has n + 1 roots. Since the degree of p ( x ) is less than n +1, p ( x ) must be the zero polynomial. Hence c 1 = c 2 = = c n +1 =0 Since the system V c = 0 has only the trivial solution, the matrix V must be nonsingular. SECTION 5 2. B = A T A = a T 1 a T 2 . . . a T n ( a 1 , a 2 ,..., a n a T 1 a 1 a T 1 a 2 a T 1 a n a T 2 a 1 a T 2 a 2 a T 2 a n . . . a T n a 1 a T n a 2 a T n a n 5. (a) 111 212 4 - 21 23 1 11 2 + - 1 - 1 (1 2 3) = 60 1 11 - 14 (c) Let A 11 = 3 5 - 4 5 4 5 3 5 A 12 = 00 A 21 = (0 0) A 22 = (1 0)

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Section 5 15 The block multiplication is performed as follows: A 11 A 12 A 21 A 22 A T 11 A T 21 A T 12 A T 22 = A 11 A T 11 + A 12 A T 12 A 11 A T 21 + A 12 A T 22 A 21 A T 11 + A 22 A T 12 A 21 A T 21 + A 22 A T 22 = 10 0 01 0 00 0 6. (a) XY T = x 1 y T 1 + x 2 y T 2 + x 3 y T 3 = 2 4 12 + 1 2 23 + 5 3 41 = 24 48 + 46 + 20 5 12 3 (b) Since y i x T i =( x i y T i ) T for j =1 , 2 , 3, the outer product expansion of YX T is just the transpose of the outer product expansion of T . Thus T = y 1 x T 1 + y 2 x T 2 + y 3 x T 3 = + 36 + 20 12 53 7. It is possible to perform both block multiplications. To see this suppose A 11 is a k × r matrix, A 12 is a k × ( n - r ) matrix, A 21 is an ( m - k ) × r matrix and A 22 is ( m - k ) × ( n - r ). It is possible to perform the block multiplication of AA T
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## This note was uploaded on 04/18/2010 for the course FINANCE 1231854365 taught by Professor Wuyiling during the Spring '10 term at Nashville State Community College.

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HW2_sol - 14 CHAPTER 1 Conversely suppose there exists a...

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