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us inverse tut- all .e'ru1‘ljlgxercise Set "troduction to Systems of Linear Equations 31 In each part, determine whether the equation is linear in x1, x2, and x3: (a) 3x1 — ﬁxz + -\'3 = 0 (b) —2.V1 — 4-\'2 + .\'2~\‘3 = 5 (C) 3n = 5A2 — 7-\'3 (d) x," — 3x2 + 4x; = 17 (e) 4.1”. +.\‘22/7 — 3.\‘3 = —I (f) mm — 7'er +1.4x3 = 14‘” . In each part, determine whether the equations form a linear system.
(a) —2.\' + 4y 2 = 2 (b) x = 4
3x — 3 = o 2.1- = 8
y
(c) 4.\'—y+22=—l (d) 3Z+.\‘=—4
—.\'+(1n2)y—33= O y+53= |
6x + 2: = 3
__\- _ )7 _ Z = 4 . In each part, determine whether the equations form a linear system. (a) M — x2 + x: = c050!)
3X] — x2 .\'3 = 2 (b) 5y + w = l
2x+5y—4z+w=1 (c) 7x, — x; + 2x; = 0 (d) x] +.\‘2 = x3 +.\'.;
2X1 + x; — X331; = 3
—.\'| + 5.\‘2 — X4 = —1 . For each system in Exercise 2 that is linear, determine whether it is consistent. . For each system in Exercise 3 that is linear, determine whether it is consistent. . Write a system of linear equations consisting of three equa- tions in three unknowns with (a) no solutions.
(b) exactly one solution. (c) inﬁnitely many solutions. . In each part, determine whether the given vector is a solution of the linear system
3N] + 2X2 — 2.1] = 1
2X] — .\'2 + ,\‘3 = 2
x] + 3x; — 3x; = —l 10. ll. 12. Chapter1 Exercise Set 75 (a) (5, —4,0)
(d) ($1? 1) In each part, determine whether the given vector is a solution
of the linear system “0 (%~ ‘75 0)
(e) (—3, 0, —5) (c) (3, —2, 2) .\‘| + 2x; — 2X3 = 3
3.1] — .\'2 + .\'3 = I
—.\‘| + 5x; — 5.\'3 = 5 (b) e, 4.0)
(d) (4. 2—0. a (e) (41%.2) In each part, ﬁnd the solution set of the linear equation by
using parameters as necessary. (a) 2x +4y = 3
(b) 3X] — 5X2 + X] + 4X4 = 9 (c) (5, 8, I) In each part, ﬁnd the solution set of the linear equation by
using parameters as necessary. (a) 3x. — 5x; + 4x; = 7
(b) 3v—8w+2x—y+4z :0 In each part, ﬁnd a system of linear equations corresponding
to the given augmented matrix. 2 5 6 3 o —2 5
(a) 012 (b) 7 1 4 —3
—1 0 0 0 —2 1 7
1 5 7 —1 3
(”[2 2 1 1 0]
1 o 0 0 7
0 1 o 0 —2
(d) 0 0 1 0 3
o 0 0 1 4 In each part, ﬁnd a system of linear equations corresponding
to the given augmented matrix. 2 —2 —3 4 0 4 2 —1 —2 (a) 3 —2 (b) [—6 0 —3 5 —2]
4 0 l 3 5 7 (c) —5 6 —1 —1
8 o 0 —2
—3 0 1 —4 —2
4 0 —4 —1 2
(d) 1 —3 o —4 0
0 3 76 Chapter 1 Systems of Linear Equations and Matrices ‘ ' 2 3 4 I 0 6 3 ——4
13. In each part, ﬁnd the augmented matrix for the given system (a) i) 1 3 3 (b) 0 1 3 7 2
of linear equations. 0 0 1 2 0 0 1 1 4
' ' —— ‘ 6x = 0
(a) ”241 = 6 (b) 3M 1‘3 + 4
3.“ = 8 2X2 -— .\'3 -— 5X4 = "2 1 2 4 g
9 ' - ——3 (c) 0 I ——3
M — 0 0 0 1
(C) 2X2 " 3X4 + .\'5 = 0 1 —3 2 O 6 1
"3-\‘1 —— x2 + x; = ——l _—
0 0 I I 2 I
6N} + 2X2 " X3 + 2X4 -— 3.\'5 = 6 (d) 0 0 0 1 2 4
(d) "" ” "3 = 4 0 0 o 0 0 0
.\‘2 + .\'4 = 9 18. In each part, suppose that the augmented matrix for a system
of linear equations has been reduced by row operations to the 14. In each part, ﬁnd the augmented matrix for the given system
given row echelon form. Solve the system. of linear equations. (a) 3.1" —— 2x; = ——I (b) 2x, + 2x; = l 1 0 0 0
4x1 + 5X2 = 3 3X1 —- x2 + 4X3 = 7 (a) O 1 0 _2
7x; + 3X2 2 2 6X1 + .\'2 -— .\’3 = 0 0 0 1 4
(c) x1 + 2x; —— x4 + x5 = l 1 0 0 3 2
3'V2+'l3+7\. —'\5:21 (b) 0 1 0 1 0
M '4 0 0 1 —4 1
(d) M .i i 2 l O 0 0 2 ——2
7 1 = 3 0 1 —2 0 0 1
3 - (c) 0 0 o 1 7 0
Gaussian Elimination 0 0 0 0 0 0
15 In each part, determine whether the matrix is in row echelon 1 2 0 0
h I f rm, both, or neither.
form, reduced row ec e on o (d) 0 0 1 0
I 0 0 I 0 0 0 0 0 0 0 0 1
(a) 0 I 0 (b) 0 O l (c) 0 0 I
0 O 1 0 0 0 0 0 0 I} In Exercises 19—22, solve the linear system by Gauss-Jordan
limination. 1:
4 I 0 3 4 e
@163] (”[0110 _ __4
0 1 1 0 19. x1 + 2x; —— 3X3 = 6 20. 2x1 + 2A3 + 2.\3 _—
0 0 2x1 —— x2 + 4x3 :1 ~23] + 5x: :14] = 111
.‘ x x =
(f) [0 1 3 :1] (g) 0 0 x1 " -\'2 + .\'3 = 3 \l + 2 3
I 0 I 0 0
21 3v——y+z+7w=l3
16. In each part, determine whether the matrix is in row echelon ——2.\' + y _ z __ 3w = __9
form, reduced row echelon form, both, or neither. ——2x + y — 7w = ——8
I I 2 0 0 I I 0 0
(a) O l l (b) 0 l I (c) 0 O l 22 __ 2). + 3.\ = 3
0 0 I I 0 I 0 O 0 3\+6y——37=——2
6\ + 6y + - = 4
I 2 3 I ' -
0 0 (e) 1 —2 2 0 it In Exercises 23—26, solve the linear system by GauSSian elim—
(d) 0 0 0 0 O 1 ination 4
0 0 O I . 24. Exercise 20 2 4 0 1 23. Exercise 19
26. Exercise 22 1
(g)i00012 17. In each part, suppose that the augmented matrix for a system
need by row operations to the -of linear equations has been red
given row echelon form. Solve the system. 25. Exercise 21 P In Exercises 27—30, determine whether the homogeneous sys—
tem has nontrivial solutions by inspection (Without pencrl and paper). 4 27. 3x. + 2X2 — X3 + 6X4 = 0
2X] -— 5X} -— X4 = 0
—6.\'1 — 21'; + 3X3 —- 3X4 = 0 28. 4x1 — 3x; — x3 = 0 29. (mm + aux; + aux; = 0
3.\'z -— 5.\'3 = 0 3X3 = 0 [1211‘] + (122.\"_i + (123.\‘3 = 0 30. 3X] -‘ 2x: = 0
6X1 — 4X2 = 0 3‘ In Exercises 31—37, solve the given linear system by any
method. '4 31.2.\‘+y+4z=0 32. 3.\'+ y— 2:0
3x+y+6z=0 -x+2.v-21=0
4x+y+9z= -\'+ y- 1: 33. X] -— x; + 7X3 + .\'4 = 0
xi + 2.\'2 — 6X3 — X4 = 0 34. v—2w+2x=0
211—— v+4w——3,\'=0
411—— v+6w—4.\‘=0
——2u+2v—6w+5x=0 35. 3w + 3x + 52 = 0
— .r+ y—3z= 2w—— .\'+3)’—- 2:
—3w+ .\'-4y+5z=0 36. .\'1 + 3x2 -— .\'4 = 0
—.\'| + 4X2 + 2X3 = O
— .\"_i— .\‘3—.\‘4=0 2X1 — 4X2 ‘l' X3 + .\'4 = 0
X] -— 2x2 -— .\'3 + .\'4 = 0 37. 411+ 312 — 213 —- [4 =0
—— 12+613—414=0 —211-— 12 + [4:0
—]1+12+13—‘ 14:0 1" In Exercises 38—41, determine the values of u for which the
system has no solutions, exactly one solution, or inﬁnitely many
solutions. «:1 38. .r+2y+ 2:2 39. x+2y+ :=2
2x—2y+3z=l 2.\'—2_v+ 3::1
.\' + 2y — a: = a x + 2_\1——(a2 - 3): = a
40. .\‘+2y—— 32: 4
3.\'— )'+ 52:: 2
4.\'+ y+(az—2)z=a+4
41. x+ ,V‘l' 7: =—7
2x+3y+ 17: =11 60 X + 2y + (a2 +1): b‘ In Exercises 42—43, solve the following systems, where a, b,
and c are constants. ‘6 Chapter 1 Exercise Set 77 42. 2x —— y = 0 43. x1 + x2 + 233 = a
3.\‘ + 2)’ = 0 2X1 + .\‘3 = b
X; + 3.\'3 = C 44. Find two different row echelon forms of i3 {ll This exercise shows that a matrix can have multiple row ech- elon forms.
45. Let
a 0 b 2
a a 4 4
0 a 2 b be the augmented matrix for a linear system. Find for what
values of a and b the system has (a) a unique solution. (b) a one-parameter solution. (c) atwo-parameter solution. (d) no solution. 46. For which value(s) of a does the following system have zero
solutions? One solution? Inﬁnitely many solutions? X] +.\'2 +.\'3 = 4
.\'3 =2 ((12 — 4).\‘3 = a — 2
47. Solve the following system of nonlinear equations for the un- known angles oz, [3, and y, where 0 S a 5 3,0 _<_ 18 < 27r,
andO 5 y f 7r. 2sina+ cosﬁ— tany=l
—4sina+ cosﬂ+ tany=0
—25ina +3cosﬁ +2tany =4 48. Solve the following system of nonlinear equations for .\', y, and 2.
2x2 + y: — 3z2 = —8
x2 —— y2 + 222 = 7
.\‘2 +2.112 —- z2 = I 49. Find positive integers that satisfy
x + _v + z = 9
x + 5y + 102 :44
50. Find values of a, b, and c such that the graph of the polyno- mial p(.\‘) = axZ + bx + c passes through the points (I, 2),
(—I, 6), and (2, 3). 51. Use Gauss—Jordan elimination to solve for .i" and y’ in terms
of x and y. 52. Use Gauss—Jordan elimination to solve forx’ and y’ in terms ofx and y. 1
x = .\" cos 9 — y' sin 9 y = x’ sin 6 + y’ c059 I' 78 Chapter 1 Systems of Linear Equations and Matrices I i ' ' ‘bl 59. Consider the matrices
I ' 53. (a) If A is a 4 x 6 matrix, what is the max1mum pos51 e 4 9 I . . f)
'I number of leading 1’s in its reduced row echelon form. 2 0 1 _7 2] C — —3 0 ’
I — B = , __ |- (b) If B is a 4 x 7 matrix whose last column has all tzerois‘; A _ [—4 6 ’ 5 3 0 2 1
' I what is the maximum pOSSlble number of 'parame ers
l I the general solution of the linear system With augmented _2 1 8 0 3 0
I I matrix B? - D = 3 0 2 , E - —5 l l I (c) If C is a 6 x 3 matrix, what is the minimum pOSSible 4 _6 3 7 6 2 in any row echelon form of C? I f ros -
4 III number Of rows 0 25 In each part compute the given expressmn (where pOSSible) l . h that
' l.‘ ‘ uired) Fmd values Of a, b’ and C suc . — E (c) 5A
I I'I 54I ifglgclegiisorfeilﬁx) = ax: + bx + c passes throngh the Pomt (a) D + E (b) D 30 I- I (—l, 0) and has a horizontal tangent at (2’ ’9)‘ (d) ——9D (e) 23 " C (f) 7E — ' " . , . . _ tr(D) 55 (a) Find a system of two linear equations 1“ the Yanﬁblgs (g) 2(D + 515) (h) B B (1)
x, y, and 2 whose solutions are given parametrica y Y (j) tr(D _ E) (k) 2 tr(4 B) (l) tr(A) x=3+t,y=ha“dz=7‘2" I rt compute the
other parametric solution to the same s Stem in ' e matrices in Exercise 59, in each pa
Y 60. Usmg th I (b) :Igiﬁrihe parameter is r, and "I = 1-. given expression (where possible). (3 ED
\9 56. Let A be a 3 x 3 matrix. Express the following sequence of (a) AB (b) BA (c) CT !I I' row operations on A in a simpler form: (d) (AB)C (e) A(BC) (f) C IiI I Add the ﬁrst row to the third row (g) (DC)T (h) (CTB)AT (i) tr(DDT)
I Subtract the third row from the ﬁrst row I
l T T t ETC 3)
I; Add the ﬁrst row to the third row (j) "(457 _ D) (k) tr(ATC +2E )(l) r(( ) Multiply the ﬁrst row by —l
lie in Exercises 61—64 the given matrix represents an augmented matrix for a linear system. Write the corresponding set of lin- I I Matrices and Matrix OperatIOnS and use Gaussian elimination to ‘ ' for the system,
' th fol- ear equations essar .
II 57. Suppose that A 3’ CI D, and E are matrices mm c solve the linear system. Introduce free parameters as nec );
lowing sizes:
I (5x6) (5x6) (6x3) (5x3) (6x5) 2 0 3 3 _
I I I ' ' trix ex ression
lI ln each part, determine whetherdthﬁe illeniilathe 512: of the 1 4 _1 2 —4 1 6
I'. is deﬁned. For those that are e n g —2 _8 2 63' —4 O 3 —1
III resulting matrix. 62. 3 12 —3 1 —1 3
I I EA 0
I' (a) BA (b) AC + D (c) B + 0 0 0
III I (d) B + AB (e) E(B + A) (f) (EA)C 3 1 —2
' (g) ATE (h) DT(A + ET) 64. _9 _3 6
. . 1
58 Suppose that A, B, C , D, and E are matrices With the fol— 6 2
I lowing sizes: 65. Let 3
I —-6
I A B C D E 3 2 _5
= 0 —-l 2
(4x1) (4x5) (5x3) (3x5) (1x4) A: 4 5 61 and B 5 5 4
I . . 2 0
I I t m'ne whether the given matrix expresswn . te to
I I In eaCh pan, de er I hat are deﬁned, give the size 0f the Use the row method or column method (as appropria ) is deﬁned. For those t . . d
l I resulting matrix. ﬁn (a) the ﬁrst row of AB (b) the third row of AB BT(ET + A)
a AE (b) ATB (c) f I
( ) + ( ) B(C + DT) (f) (E B)T + CD (c) the second column of AB (d) the ﬁrst column 0 A
B D e ' BA
I III (d) 3 T h EA + DC (e) the third row of AA (f) the third column of A
II In) I (DB ) ( ) 66. Referring to the matrices in Exercise 65, use the row method
or column method (as appropriate) to ﬁnd (a) the ﬁrst column of AB. (c) the second row of BB. (d) the ﬁrst column of AA. (e) the third column of AB. (f) the ﬁrst row of BA. 67. Referring to the matrices in Exercise 65 and Example 9 of
Section 1.3, (a) express each column vector of AA as a linear combina-
tion of the column vectors of A. (b) express each column vector of BB as a linear combina-
tion of the column vectors of B. 68. Referring to the matrices in Exercise 65 and Example 9 of
Section 1.3, (a) express each column vector of AB as a linear combina-
tion ofthe column vectors of A. (b) express each column vector of BA as a linear combina-
tion of the column vectors of B. 69. In each part, ﬁnd matrices A, x, and b that express the
given system oflinear equations as a single matrix equation
Ax = b, and write out this matrix equation. (a) 5x + y + z = 2 (b) .\'1 + x2 — x; —— 7x4 = 6
2X +3Z=l — .\'2+4.\'3+ .\‘4=l
.r + 2y = 0 4x. + 2x; + x3 + 8n = 0 70. In each part, ﬁnd matrices A, x, and b that express the
given system of linear equations as a single matrix equation
Ax = b, and write out this matrix equation. (a) 2x1 — x; + 3x; = 4
.\'1 + 3X1 = —2
2x2 — x; = l
—.\‘1 + 2X3 = 0
(b) 4x, + 4x; + 4x; = 4
—2.\‘2 —‘ 3X2 — .\'3 = 0
4.x"; — 2x; = —2 71. In each part, express the matrix equation as a system of linear
equations. 5 6 —7 x. 2
(a) —l —2 3 x2 = 0
0 4 ——l x; 3 1 l l x, 2
(b) 2 3 0 x2 = 2
5 —3 —6 x3 —9 equations.
. I 4 0 —l .\'| 3
' (a) 3 2 4 x: = 0
‘2 l 0 .\‘3 —2 (b) the third column of BB. '72- In each part, express the matrix equation as a system of linear (b)—-213—2y= Chapter1 Exercise Set 79 4 —l l 3 w
4 —l 0 2 x 0000 2—5 —l—6 z D“ In Exercises 73—74, ﬁnd all values of k, if any, that satisfy the equation. 4
l l 0 k
73. [k l i] l 0 2 1 =0
0 2 —3 l
l 2 0 3
74. [3 3 k] l l 2 3
4 0 l k D- in Exercises 75—76, solve the matrix equation fora, b, c, and d. 75. 76. 77. 78. 79. 80. 81. 3 a _ b 0—2:!
1 (1+1) — c+2d 0 [1—17 b+a _ 9 3
4d+c 2d—2c — 7 6 Let A be any m x 11 matrix and let 0 be the m x n matrix
each of whose entries is zero. Show that if M = 0, either
k = 0 or A = 0. Show that if a square matrix A satisﬁes
A3+4A2—2A+71=0
then so does AT. Prove: If A is an m X n matrix and B is the n x 1 matrix
each of whose entries is 1/", then 7i
7‘2
AB =
7”I
where F, is the average of the entries in the ith row of A. (a) Show that if B is any matrix with a column of zeros and
A is any matrix for which AB is deﬁned, then AB also has a column of zeros. (b) Find a similar result involving a row of zeros. Find the 4 x 4 matrix A = [aij] whose entries satisfy the
stated condition. (a) av =i—j (b) a.) = (4)117
_ 0 Ii—jlzi
(c) ""_[—i li—j|<l 80 Chapter 1 Systems of Linear Equations and Matrices s in Exercise 87, verify that (b) (A+C)T =AT+CT 89. Using the matrices and scalar 82. Consider the function y = f(x) deﬁned for 2 x 1 matr1ces X by y := Ax, where
1 l
A = 0 1 ' - ‘ 10
t .‘ to ether w1th .\ 1n each case be - . ‘4
1):): dfe(s\c)ribegthe action of f? of the followmg matr1ces. 2 6 3
(a) x = (11) (b) x = (o) 90. A =1: :1 9" B = [—5 4]
2
4 . _
(‘9 "’ = (3) (d) A — (—2) 14 91
92. — wi and columnj — 1 3 (a) (3T)T = B (c) (bA)T = 12.47 (d) (CA)T = ATCT 5 to compute the inverses w. How would 1‘? 1n Exercises 90—93,useTheorem 1.4. 83. Let 1 be the n x 11 matrix whose entry in ro
. 94. Find the inverse of IS . . .
1 1f 1 = _]
0 1f '35]- [%(€r __ e—r) %(er +e-x)]
- l .r ,—.r l ,x _ —.r
Show that A] = [A = A for every n x n matr1x A. 2(e + L ) 2(c e )
- - T —1 =
84' HOW many 3 X 3 matrices A can you ﬁnd SUCh that 95. Use the matrix C in Exerc15e 92 to ver1fy that (A )
x xy (AH)?
A .V : )’ 96 Use the matrices A and B in Exercises 90 and 91 to verify 2 Z that (A3)" = B"A"~ for all choices of .1", y, and 2? (Note that A may also depend \ and z) 97 Use the matrices A, B, and C in Exercises 90—92 to verify
on .1', _’, ~ - — —1 —1
that (ABC)‘l = C lB A .
85. If A and B are 11 x n matrices, then . ' ' f mation to ﬁnd A. 4
(a) tr(cA) = c tr(A) where c is a real number, use the g'Ve" 1n or 4 2
99. (5A) ‘= [1 3] —1 3
—5 1 _ _
100. (3AT)"=[__9 2] 101. (1+2A) '_[ 4 5] E'- ln Exercises 98—101, (b) tr(AB) =tr(BA). 2 _1
. . -1 _
86. Show that there are no 2 x 2 matr1ces A and B w1th 98. A _ [3 4]
AB — BA equal to the 2 x 2 identity matr1x l . [Hints use the previous exercise] Inverses; Algebraic Properties of Matrices ow should the coefﬁcients a, b, and c be chosen so that the 87. Let 102- 1-1
2 ——1 3 8 —3 ——5 system
A: 0 4 5 , B: 0 l 2, (Ix+by——3:=——3
—2 1 4 4 —7 6 —2.\‘ — by + cz = ——l
2 3 a.\'+3y—cz=—3
0 __
C: 7 i “:4’ b:_7 hasthesolutionx:l,y=—l,andz=2?
3 5 9 1
103. Let A be the matr1x
Show that 1 #1
(a)A+(B+C)=(A+B)+C [#2 3] (b) (AB)C = A(BC) (6) (11+ b)C 2 [1C + bC (d) 0(3 __ C) =aB ——aC matrices and scalars in Exercise 87, ver1fy that In each part, compute the given quantity. (a) A3 (b) 14—}
(d) p(A), where p(x) = x —— 2
(e) p(A), where p(.\') = 2x2 —— x +1 (f) p(A), where p(x) = x3 — 2x + 4 88. Using the
(a) a(BC) = (aB)C = B(aC) (b) A(B——C)=AB—AC (c) (B+C)A=BA+CA IJ\ ,./l.f‘\ — {nlﬂr (c) A2—2A+I 104. Repeat Exercise 103 for the matrix 3 0 —l
A: 0 —2 0
5 0 2 105. Repeat Exercise 103 for the matrix 3 0 0
A: 0 —l 3
0 —3 —l 106. Let pl(x) = .1‘2 — 9, p3(.\‘) = .r + 3, and p3(.\') = x — 3.
Show that p1(A) = p3(A)p3(A) for the matrix A in Exer—
cise 105. 107. Show that if p(.\') = .r2 _ (u + d).\‘ + (ad — be) and then p(A) = 0. 108. Show that if p(.\') = .\'3 — (a + b + c)x2 + (ab + ae +
be — cd).\' — (1(be — Cd) and a 0 0
A = 0 b c
0 d e
then p(A) = 0.
109. Consider the matrix
[I11 0 0
O (123 0
A = .
0 0 . - - (1",, where (man ~ - - (1,," 9e 0. Show that A is invertible and ﬁnd
its inverse. 110. Show that ifa square matrix A satisﬁes the equation
A2 + 5A — 21 = 0, then A" = 1m + 51). 111. (a) Show that a matrix with a row of zeros cannot have an
inverse.
(b) Show that a matrix with a column of zeros cannot have an inverse. 112. Assuming that all matrices are n X n and invertible, solve
for D. ABCTDBATC = A87 113. Assuming that all matrices are n X n and invertible, solve
for D. CTB"AZBAC"DA‘ZBTC‘1 = CT 114. If A is a square matrix and n is a positive integer, is it true
that (A")T = (AT)"? Justify your answer. Chapter 1 Exercise Set 81 115. Simplify:
D"CBA(BA)"C ‘(C"'D)’l
b In Exercises 116—117, determine whether A is invertible, and if so, ﬁnd the inverse. [Hint Solve AX = l for X by equating
corresponding entries on the two sides.] 63 1 1 1 l 0 l
116. A: O 0 l 117. A: O l 0
1 l 0 1 0 w1 D In Exercises 118—121, use the method of Example 8 of Sec-
tion 1.4 to ﬁnd the unique solution of the given linear system. 4 118. 331+ 2x; :1 119. .\'1+3.X2 = 0
4x1 — 5X2 2 2 2X1 — 5X1 = 3
120. 7x1+ 2x; = 3 121. 3.\'1 — 2x2 = 6
3X1 + .\‘2 = 0 —-\'| +4.\'2 = 1 122. Prove: If B is invertible, then AB‘1 = B"A if and only if
AB = BA. 123. Prove: lfA is invertible, then A + B and 1 + BA“l are both
invertible or both not invertible. 124. Find a matrix K such that AKB = C given that 14
200
A=—2 3,3: ,
01—1
1—2
86—6
C=6—ll,
—400 125. (a) Show that if A, B, and A + B are invertible matrices
with the same size, then A(A“ + B“)B(A + B) l= I (b) What does the result in part (a) tell you about the matrix
A“ + B"? 126. A square matrix A is said to be idempotent if A2 = A.
(a) Show that ifA is idempotent, then so is l — A. (b) Show that if A is idempotent, then 2A — 1 is invertible
and is its own inverse. 127. Show that ifA is a square matrix such that Ak = 0 for some
positive integer k, then the matrix A is invertible and (l—A)" =1+A+A3+...+Ak-1
Elementary Matrices and a Method for Finding A"1 128. Decide whether each matrix below is an elementary matrix. (a) [f (1,] (b) [2, ﬂ 82 Chapter 1 Systems of Lin 1 0 0
(c) 1 0 0 (d)
0 1 0 129. Decide whet
' ﬂ 0
(”I 0 1
l 0 0
(c) ——2 l O
0 0 l l- ' 130. Find a row operation
trix that will restore t ' tity matrix. '5- 1 [1 0]
'_ .- (a) 0 ﬂ ' ' 1 0 0
(c) 0 0 l
0 1 0 trix that will restore t
: tity matrix. 132. In each part, an elementa
given. Write down the ro II and show that the product
i operation to A. 1 0 0
(b) E = 0 1 —3
0 0 1 —2 1 0 A = 1 —3 0 ear Equations and Matrices
x E and a matrix A are tion corresponding to E
lying the row rt, an elementary matri
ow opera
EA results from app 133. In each pa
given. Write down the r and show that the product
operation to A. her each matrix below is an elementary —24
—37 A: 5, use the following matrices. Ponding elementary ma- 1:- In Exercises 134—13 and the corres .
o the 1den— he given elementary matrix t 190
(b)010
001 onding elementary ma- I ' 131. Find a row operation and the corresp he iden- he given elementary matrix to t S the equation- nd an elementary matrix E that satisﬁe
(b) EB = A 134. F1
(a) EA = B (c) EA=C (d) EC=A x E that satisﬁes the equation.
(b) ED = B
(d) EF = B 1 135. Find an elementary matri
-2 = D
0 (c) EB = F use the inversion algorithm to ﬁnd the P In Exercises 136—150, .
e ex1sts. inverse of the given matrix, if the invers 137. [F ry matrix E and a matrix A are
w operation corresponding to E EA results from applying the row
3 6 m1 1 m1 ' 0 1 _
'w=111 A-1 141. 1N l m1:— m1— v.1-
|— El- u. -I 0 —l
143. 144. 0 l l
l 1 0 1 () 0
‘45- 0 3J5 J? 146.
0 ﬂ —4J§ U-I— U-I-- VI...

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- Spring '14
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