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**Unformatted text preview: **6 —1 1 O O 10 O O
6. The coefﬁcient matrixA= 1 O O —1 2 has reducedformAR= O 1 —1 O
1 0 O O —2 O O O
O 2
1 12
With x3=a,x5=ﬁthe general solution can be writtenasX=a 1 +ﬁ 0 so the
O 4
O 1
solution space of AX =Ohas dimension2=m— rank (A)=5—3.
8 O —2 O O 1
. . 2 —1 O 3 O —1
8. The coeﬂicIent matrIXA— 0 1 1 O _2 _1 has reduced form
0 O 0 1—3 2
1 O O O 7/6 —5/4
_ O 1 O 0 —20/3 9/2 . _ _ .
AR— 0 O 1 0 14/3 _11/2 .W1th 235—01,:35—Bthe general solutlon can be
0 O O 1 —3 2
—7/6 5/4
20/3 —9/2
writtenas X =a _1:/3 +ﬁ 1y; ,so the solution space of AX =0has dimension
1 O
0 1
2=m— rank (A)=6'—4.
4 —3 O 1 1 —3
. . O 2 O 4 —1 —6
10. The c0efﬁc1ent matnx A— 3 _2 O O 4 _1 has reduced form
1; 2 1 —3 4 0 O
1 0 0 0 *W/s “V;
0 1 0 0 to V . .
AR : O O 1 O. “AD 4‘: . Wlth x5 = 04,235 = B the general solut10n can be
0 O O 1 47/“, 46/5
'95”: V:
“Wk, .18,
itte asX “V” + V; th lt' fAX Oh d" ‘
wr n :04 ,so es0u10nsa =
27/“) 9/5 p ce 0 as 1mens10n
1 0
O 1
=m—rank(A)=6—4
2 O 0 O —4 0 1 1
0 2 O O 0 —1 1 —1
12. The coefﬁcient matrixA= O O 1 —4 O O O 1 has reduced form
0 1 —1 1 O O O O
01 O O —1 1 —1 O
1 O O O O —3 7/2 —1/2
0 1 O O O —1/2 1/2 —1/2
AR: 0 0 1 0 O —2/3 2/3 —1 . With 335 =a,:1:7 =B,xg =7the general
0 O O 1 O —1/6 1/6 —1/2
0 n n n1_q/o '2/0 .1/0 (min/meal) 3 —7/2 1/2
1/2 —1/2 1/2
2/3 —2/3 1
solution can be written as X = a + ,3 + 7 , so the solution
1 0 0
0 1 0
0 0 1 space of AX = 0 has dimension 3 = m— rank (A) = 8 — 5. Section 7.6 6. The solution Space of AX = 0 has dimension 2 , as shown by the two arbitrary parameters
in the general solution. 8. The solution space of AX = 0 has dimension 2, as shown by the two arbitrary parameters
in the general solution. 10. The solution space of AX = 0 has dimension 2, as shown by the two arbitrary parameters
in the general solution. 12. The solution space of AX = O has dimension 3, as shown by the three arbitrary parameters
in the general solution. Section 7.7 4—231051 2. The augmented matrix is 1 0 0 _3 f 8 . The reduced form of this matrix is 2—301316 1 0 0 —3 E 8
0 1 0 _7/3 3 0 . Since rank (A) = rank (AEB) the system has solutions which
0 0 1 52/9 3 —31/3
8 3
can be expressed as X = 0 + a 7/3 where a is arbitrary
—31/3 —52/ 9 ' 0 1 j 2 0 —3 E 1
8. The augmented matrix is 1 _1 1 g 1 . The reduced form of this matrix is
2 —4 1_ 5 2
1 0 0 3/4
0 1 0 _1/12 . Since rank (A) = rank (AB) = number of unknowns = 3, the
0 0 1 1/6
3/4
system has a unique solution X = —1 / 12 1/6 3—23—1 10. The augmented matrix is t
4 3 1 4 ). The reduced form of this matrix is < 1 0 3 5/17 . Since rank (A) = rank (AB) = number of unknowns = 2, the system
0 1 S 16/17 . . _ 5/17
has a unique solution X — < 16/17 12. The augmented matrix is 2 _6 1 g _5 . The reduced form of this matrix is 1 0 0 E ——137/48
0 1 0 f 1/6 . Since rank (A) 2 rank (A53) 2 number of unknowns = 3, the
0 0 1 S 41/24
—l37/48
system has a unique solution X = 1/6
41/24
——6 2 -1 1 E 0 14. The augmented matrix is 1 4 0 _1 _5 . The reduCed form of this matrix 1 1 1 -—7 E 0
is 1 0 0 21/23 S -—15/23
0 l 0 —ll/23 3 —25/23 . Since rank (A) 2 rank (ASE) the System has solutions
0 0 1 —171/23 3 40/23
-15/23 —21/23
which can be expressed X = ‘25/23 11/23 40/23 + a 171/23 where a is arbitrary.
0 l ...

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