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mat67-Homework5_Solutions

# mat67-Homework5_Solutions - MAT067 University of California...

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MAT067 University of California, Davis Winter 2007 Solutions to Homework Set 5 1. Suppose that A , B , C , D , and E are matrices over F having the following sizes: A is 4 × 5 , B is 4 × 5 , C is 5 × 2 , D is 4 × 2 , E is 5 × 4 . Determine whether the following matrix expressions are defined, and, for those that are defined, determine the size of the resulting matrix. (a) BA (b) AC + D (c) AE + B (d) AB + B (e) E ( A + B ) (f) E ( AC ) Results : (a) DNE (b) 4 × 2 (c) DNE (d) DNE (e) 5 × 5 (f) 5 × 2 2. Suppose that A , B , C , D , and E are the following matrices: A = 3 0 1 2 1 1 , B = bracketleftbigg 4 1 0 2 bracketrightbigg , C = bracketleftbigg 1 4 2 3 1 5 bracketrightbigg , D = 1 5 2 1 0 1 3 2 4 , E = 6 1 3 1 1 2 4 1 3 . Determine whether the following matrix expressions are defined, and, for those that are defined, compute the resulting matrix. (a) D + E (b) D E (c) 5 A (d) 7 C (e) 2 B C (f) 2 E 2 D (g) 3( D + 2 E ) (h) A A (i) AB (j) BA (k) (3 E ) D (l) ( AB ) C (m) A ( BC ) (n) (4 B ) C + 2 B (o) D 3 E (p) CA + 2 E (q) 4 E D (r) DD Results : (a) 7 6 5 2 1 3 7 3 7 (b) 5 4 1 0 1 1 1 1 1 (c) 15 0 5 10 5 5 (d) bracketleftbigg 7 28 14 21 7 35 bracketrightbigg (e) DNE (f) 10 8 2 0 2 2 2 2 2 (g) 39 21 24 9 6 15 33 12 30 (h) 0 0 0 0 0 0 (i) 12 3 4 5 4 1

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2 Results (continued): (j) DNE (k) 42 108 75 12 3 21 36 78 63 (l) 3 45 9 11 11 17 7 17 13 (m) 3 45 9 11 11 17 7 17 13 (n) DNE (o) 17 2 7 2 3 5 9 1 5 (p) DNE (q) 23 1 10 3 4 7 13 2 8 (r) 2 9 15 2 3 2 13 23 24 3. Suppose that A , B , and C are the following matrices and that a = 4 and b = 7. A = 1 5 2 1 0 1 3 2 4 , B = 6 1 3 1 1 2 4 1 3 , and C = 1 5 2 1 0 1 3 2 4 . Verify computationally that (a) A + ( B + C ) = ( A + B ) + C (b) ( AB ) C = A ( BC ) (c) ( a + b ) C = aC + bC (d) a ( B C ) = aB aC (e) a ( BC ) = ( aB ) C = B ( aC ) (f) A ( B C ) = AB AC (g) ( B + C ) A = BA + CA (h) a ( bC ) = ( ab ) C (i) B C = C + B 4. Suppose that A is the matrix A = bracketleftbigg 3 1 2 1 bracketrightbigg . Compute p ( A ) where p ( x ) is given by (a) p ( x ) = x 2 (b) p ( x ) = 2 x 2 x + 1 (c) p ( x ) = x 3 2 x + 4 (d) p ( x ) = x 2 4 x + 1 Results : (a) bracketleftbigg 1 1 2 1 bracketrightbigg (b) bracketleftbigg 20 7 14 6 bracketrightbigg (c) bracketleftbigg 39 13 26 13 bracketrightbigg (d) bracketleftbigg 0 0 0 0 bracketrightbigg 5. In each of the following, find matrices A , x , and b such that the given system of linear equations can be expressed as the single matrix equation Ax = b . (a) 2 x 1 3 x 2 + 5 x 3 = 7 9 x 1 x 2 + x 3 = 1 x 1 + 5 x 2 + 4 x 3 = 0 (b) 4 x 1 3 x 3 + x 4 = 1 5 x 1 + x 2 8 x 4 = 3 2 x 1 5 x 2 + 9 x 3 x 4 = 0 3 x 2 x 3 + 7 x 4 = 2
3 Results : (a) A = 2 3 5 9 1 1 1 5 4 , x = x 1 x 2 x 3 , and b = 7 1 0 .

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mat67-Homework5_Solutions - MAT067 University of California...

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