This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 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 1 1 1 1 1 (c) 15 5 10 5 5 (d) bracketleftbigg 7 28 14 21 7 35 bracketrightbigg (e) DNE (f) 10 8 2 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 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...
View
Full
Document
This homework help was uploaded on 04/08/2008 for the course MATH 67 taught by Professor Schilling during the Fall '07 term at UC Davis.
 Fall '07
 Schilling
 Linear Algebra, Algebra, Matrices

Click to edit the document details