ISM-Chp_2

Linear Algebra and Its Applications (3rd Edition)

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83 2.1 SOLUTIONS Notes : The definition here of a matrix product AB gives the proper view of AB for nearly all matrix calculations. (The dual fact about the rows of A and the rows of AB is seldom needed, mainly because vectors here are usually written as columns.) I assign Exercise 13 and most of Exercises 17–22 to reinforce the definition of AB . Exercises 23 and 24 are used in the proof of the Invertible Matrix Theorem, in Section 2.3. Exercises 23–25 are mentioned in a footnote in Section 2.2. A class discussion of the solutions of Exercises 23–25 can provide a transition to Section 2.2. Or, these exercises could be assigned after starting Section 2.2. Exercises 27 and 28 are optional, but they are mentioned in Example 4 of Section 2.4. Outer products also appear in Exercises 31–34 of Section 4.6 and in the spectral decomposition of a symmetric matrix, in Section 7.1. Exercises 29–33 provide good training for mathematics majors. 1 . 201 402 2( 2 ) 452 8 1 04 A −−  −= =   . Next, use B – 2 A = B + (–2 A ): 751 4 02 353 2 143 8 1 767 BA + = The product AC is not defined because the number of columns of A does not match the number of rows of C . 1 2 3 5 13 2 ( 1 ) 15 24 1 13 2 1 1 4 23 1 (1 ) 25 14 7 6 CD ⋅+ − ⋅+⋅ == = + + . For mental computation, the row-column rule is probably easier to use than the definition. 2 . 2 0 1 7 5 1 2 14 0 10 1 2 16 10 1 22 4 25 82 6 61 34 AB + +− += + = = + The expression 3 C E is not defined because 3 C has 2 columns and – E has only 1 column. 1 2 7 5 1 17 21 1 (5 ) 2 (4 ) 11 2 (3 ) 9 1 3 5 2 1 1 4 3 27 11 2 ) 1 ) 21 1 ) 1 3 6 5 CB −⋅ + + + = + + + The product EB is not defined because the number of columns of E does not match the number of rows of R .

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84 CHAPTER 2 • Matrix Algebra 3 . 2 30 4 1 340( 1 ) 11 3 03 5 2 053( 2 ) 55 IA −−  −= = =   22 411 23 (3 ) 3( ) 3 521 56 IA IA == = , or 2 304 1 3 403 ( 1 )0 1 2 3 (3 ) 035 2 03 503 ( 2 ) 1 5 6 −⋅ +− + = −+ + 4 . 3 9135 0 0 413 58 7 6 0 5 0 8 2 6 4180 0 5 AI = = 33 913 4 551 5 (5 ) 5( ) 5 5 8 7 6 40 35 30 418 2 054 0 A = = , or 3 500 9 1 3 ( 5 ) 0 5 0876 005 4 1 8 =− 5900 5 (1 )00 5300 4 5 5 1 5 05 (8 )0 0570 05 (6 )0 4 5 3 0 005 (4 ) 1 0058 2 0 5 4 0 ⋅++ − ++ =+−+ + ⋅+ +−+= ++ − ++⋅ 5 . a .
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ISM-Chp_2 - 2.1 SOLUTIONS Notes: The definition here of a...

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