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# SSM-Ch07 - Chap 7 Sec 7.1 Linear Algebra Matrices Vectors...

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Chap . 7 Linear Algebra : Matrices , Vectors , Determinants , Linear Systems Sec . 7 . 1 Matrices , Vectors : Addition and Scalar Multiplication Problem Set 7 . 1 . Page 277 5 . Matrix addition , scalar multiplication . Calculate 3 C u2212 8 D as follows. First multiply C by 3, then D by 8. This gives 3 C u003d 3 0 2 2 4 1 3 u003d 0 6 6 12 3 9 and 8 D u003d 8 6 1 u2212 4 7 u2212 8 3 u003d 48 8 u2212 32 56 u2212 64 24 . The resulting matrices have the same size as the given ones, namely 3 u00d7 2 (3 rows, 2 columns) because scalar multiplication does not alter the size of a matrix. Hence the operations of addition and subtraction are defined for these matrices, and you obtain the result by subtracting the entries of 8 D from the corresponding ones of 3 C , that is, 3 C u2212 8 D u003d 0 u2212 48 6 u2212 8 6 u002b 32 12 u2212 56 3 u002b 64 9 u2212 24 u003d u2212 48 u2212 2 38 u2212 44 67 u2212 15 . Matrix addition and scalar multiplication have properties quite similar to those of the addition and multiplication of numbers – note well that we are presently concerned with multiplying the entries of a

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Chap. 7 Linear Algebra: Matrices, Vectors, Determinants, Linear Systems 67 matrix by a number (the same number), and we are not yet multiplying a matrix by another matrix; this will be done in the next section. The next task in the problem illustrates formula (4c) on p. 276, showing that you get the same result by multiplying a matrix by 3 and the result then by 4, or by multiplying the matrix at once by 3 u2219 4 u003d 12 : 4 ue0a2 3 A ue0a3 u003d ue0a2 4 u2219 3 ue0a3 A u003d 12 A u003d 12 3 0 4 u2212 1 2 2 6 5 u2212 4 u003d 36 0 48 u2212 12 24 24 72 60 u2212 48 Finally, B u2212 1 10 A is also defined since both A and B are square matrices, of size 3 u00d7 3, and so is 1 10 A . You obtain B u2212 1 10 A u003d 0 u2212 5 u2212 3 u2212 5 2 4 u2212 3 4 0 u2212 0.3 0.0 0.4 u2212 0.1 0.2 0.2 0.6 0.5 u2212 0.4 u003d u2212 0.3 u2212 5.0 u2212 3.4 u2212 4.9 1.8 3.8 u2212 3.6 3.5 0.4 . 7 . Vectors are special matrices, having a single row or a single column, and operations with them are the same as for general matrices (and somewhat simpler). u and v are column vectors, and they have the same number of components, they are of the same size 3 u00d7 1. Hence they can be added. You obtain in this problem 33 u u003d 33 2 0 u2212 1 u003d 66 0 u2212 33 . For the next two tasks you obtain the same result because 4 v u002b 9 4 u u003d 4 v u002b 9 u u003d 4 u2212 4.5 0.8 1.2 u002b 9 2 0 u2212 1 u003d u2212 18 3.2 4.8 u002b 18 0 u2212 9 u003d 0 3.2 u2212 4.2 . Finally, u u2212 v u003d 2 0 u2212 1 u2212 u2212 4.5 0.8 1.2 u003d 6.5 u2212 0.8 u2212 2.2 . 15 . Proof of ( 3a ). A and B are assumed to be general 3 u00d7 2 matrices. Hence you have A u003d a 11 a 12 a 21 a 22 a 31 a 32 and B u003d b 11 b 12 b 21 b 22 b 31 b 32 . Now use the definition of matrix addition to obtain A u002b B u003d a 11 u002b b 11 a 12 u002b b 12 a 21 u002b b 21 a 22 u002b b 22 a 31 u002b b 31 a 32 u002b b 32 and
68 Linear Algebra, Vector Calculus Part B B u002b A u003d b 11 u002b a 11 b 12 u002b a 12 b 21 u002b a 21 b 22 u002b a 22 b 31 u002b a 31 b 32 u002b a 32 . Now remember what you want to prove. You want to prove that A u002b B u003d B u002b A .

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SSM-Ch07 - Chap 7 Sec 7.1 Linear Algebra Matrices Vectors...

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