Introduction to Linear algebra-Strang-Solutions-Manual_ver13

# 38 3 ab c ac is the same as ab c c d distributive law

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Unformatted text preview: shows 0 1 23 11 01      11 10 10 11 that M D . 01 11 11 01 2 3 2 3 2 1 1 1 6 1=2 1 7 60 7 60 1 1 7, E32 D 6 7, E43 D 6 31 E21 D 6 40 5 4 0 2=3 1 5 40 0 1 01 0 001 0 0 01 0 0 3=4 1 2 3 1 6 1=2 7 1 7 E43 E32 E21 D 6 4 1=3 2=3 1 5 1=4 2=4 3=4 1 Problem Set 2.4, page 75 1 If all entries of A; B; C; D are 1, then BA D 3 ones.5/ is 5 by 5; AB D 5 ones.3/ is 3 by 3; ABD D 15 ones.3; 1/ is 3 by 1. DBA and A.B C C / are not deﬁned. 2 (a) A (column 3 of B ) (b) (Row 1 of A) B (c) (Row 3 of A)(column 4 of B ) (d) (Row 1 of C )D (column 1 of E ).   38 3 AB C AC is the same as A.B C C / D . (Distributive law). 69   00 4 A.BC / D .AB/C by the associative law. In this example both answers are 00 from column 1 of AB and row 2 of C (multiply columns times rows).       n  1 2b 1 nb 44 2 2n 5 (a) A2 D and An D . (b) A2 D and An D . 01 01 00 00     10 4 16 2 2 2 2 2 2 6 .A C B/ D D A C AB C BA C B . But A C 2AB C B D . 66 30 7 (a) True (b) False (c) True (d) False: usually .AB/2 ¤ A2 B 2 . 3 7 7, 5 Solutions to Exercises 15 8 The rows of DA are 3 (row 1 of A) and 5 (row 2 of A). Both rows of EA are row 2 of A. 9 10 11 12 13 14 15 16 17 The columns of AD are 3 (column 1 of A) and 5 (column 2 of A). The ﬁrst column of AE is zero, the second is column 1 of A C column 2 of A. &quot; # a aCb AF D and E.AF / equals .EA/F because matrix multiplication is c cCd associative. &quot; # &quot; # aCc bCd aCc bCd FA D and then E.FA/ D . E.FA/ is not c d a C 2c b C 2d the same as F .EA/ because multiplication is not commutative. &quot; # 001 (a) B D 4I (b) B D 0 (c) B D 0 1 0 (d) Every row of B is 1; 0; 0. 100 &quot; # &quot; # a0 ab AB D D BA D gives b D c D 0. Then AC D CA gives c0 00 a D d . The only matrices that commute with B and C (and all other matrices) are multiples of I : A D aI . .A B /2 D .B A/2 D A.A B / B .A B / D A2 AB BA C B 2 . In a typical case (when AB ¤ BA) the matrix A2 2AB C B 2 is different from .A B /2 . (a) True (A2 is only deﬁned when A is square) (b) False (if A is m by n and B is n by m, then AB is m by m and BA is n by n). (c) True (d) False (take B D 0). (a) mn (use every entry of A) (b) mnp D ppart (a) (c) n3 (n2 dot products). (a) Use only column 2 of B (b) Use only row 2 of A (c)–(d) Use row 2 of ﬁrst A. 3 3 2 2 111 1 1 1 7 7 6 6 1 1 5 has aij D . 1/i Cj D A D 4 1 2 2 5 has aij D min.i; j /. A D 4 1 1 23 2 1=1 1=2 1=3 13 1 1 6 7 “alternating sign matrix”. A D 4 2=1 2=2 2=3 5 has aij D i=j (this will be an 3=1 3=2 3=3 example of a rank one matrix). 18 Diagonal matrix, lower triangular, symmetric, all rows equal. Zero matrix ﬁts all four. a31 19 (a) a11 (b) `31 D a31 =a11 (c) a32 . a11 /a12 (d) a22 . a21 /a12 . a11 2 3 2 3 0040 0008 600047 600007 7 6 7 6 20 A2 D 6 7 ; A4 D zero matrix for strictly triangular A. 7 ; A3 D 6 400005 400005 0000 2 x 6y 6 Then Av D A 6 4z t 3 2 2y 0000 3 2 4z 3 2 8t 3 7 6 2z 7 6 4t 7 607 76 7 6 7 6 7 7D6 7 ; A2 v D 6 7 ; A3 v D 6 7 ; A4 v D 0 . 5 4 2t 5 405 405 0 0 0 Solutions to Exercises 16 2 3 21 A D A D A D    D  &quot; :5 :5 :5 :5 # &quot; :5 :5 # and .AB/2 D z...
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## This note was uploaded on 09/25/2012 for the course PHY 103 taught by Professor Minki during the Spring '12 term at Korea Advanced Institute of Science and Technology.

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