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hw2solutions - Math 215 HW #2 Solutions 1. Problem 1.4.6....

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Unformatted text preview: Math 215 HW #2 Solutions 1. Problem 1.4.6. Write down the 2 by 2 matrices A and B that have entries a ij = i + j and b ij = (- 1) i + j . Multiply them to find AB and BA . Solution: Since a ij indicates the entry in A which is in the i th row and in the j th column, we see that A = 2 3 3 4 . Likewise, B = 1- 1- 1 1 . Therefore, AB = 2 3 3 4 1- 1- 1 1 = 2 1 + 3 (- 1) 2 (- 1) + 3 1 3 1 + 4 (- 1) 3 (- 1) + 4 1 =- 1 1- 1 1 . Also, BA = 1- 1- 1 1 2 3 3 4 = 1 2 + (- 1) 3 1 3 + (- 1) 4 (- 1) 2 + 1 3 (- 1) 3 + 1 4 =- 1- 1 1 1 . 2. Problem 1.4.16. Let x be the column vector (1 , , . . . , 0). Show that the rule ( AB ) x = A ( Bx ) forces the first column of AB to equal A times the first column of B . Solution: Suppose that x has n components. Then, in order for Bx to make sense, B must be an m n matrix for some m . In turn, for the matrix product AB to make sense, A must be an ` m matrix for some ` . Now, suppose A = a 11 a 1 m . . . . . . a ` 1 a `m and B = b 11 b 1 n . . . . . . b m 1 b mn . Then AB = a 11 a 1 m . . . . . . a ` 1 a `m b 11 b 1 n . . . . . . b m 1 b mn = m i =1 a 1 i b i 1 ] m i =1 a 1 i b in . . . . . . m i =1 a `i b i 1 m i =1 a `i b in . Hence, ( AB ) x = m i =1 a 1 i b i 1 ] m i =1 a 1 i b in . . . . . . m i =1 a `i b i 1 m i =1 a `i b in 1 . . . = m i =1 a 1 i b i 1 m i =1 a 2 i b i 1 . . . m i =1 a `i b i 1 , 1 which is just a copy of the first column of AB . On the other hand, Bx = b 11 b 1 n . . . . . . b m 1 b mn 1 . . . = b 11 b 21 . . . b m 1 , which is the first column of B . Therefore, A ( Bx ) is A times the first column of B ; since A ( Bx ) = ( AB ) x and ( AB ) x is the first column of AB , we see that the first column of AB must be A times the first column of B . Though its not part of the assigned problem, the same argument with different choices of x (e.g. (0 , 1 , , . . . , 0), etc.) will demonstrate that each column of AB must be equal to A times the corresponding column in B . 3. Problem 1.4.20. The matrix that rotates the xy-plane by an angle is A ( ) = cos - sin sin cos . Verify that A ( 1 ) A ( 2 ) = A ( 1 + 2 ) from the identities for cos( 1 + 2 ) and sin( 1 + 2 )....
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hw2solutions - Math 215 HW #2 Solutions 1. Problem 1.4.6....

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