We expand by minors about row 1 to find D.
Expand by minors about row 1.
Verify that and Thus,
Check that the solution set is NOW TRY
Determining When Cramer’s Rule Does Not ApplyUse Cramer’s rule, if possible, to solve the system.
First, find D.
Since here, Cramer’s rule does not apply and we must use another method to
solve the system. Multiplying each side of the first equation by 3 shows that the first
two equations have the same solution set, so this system has dependent equations and
an infinite solution set. NOW TRY
D= 0
= 0
= 2132 - 6112 + 1102
= 2 `
- 9
2
12
- 3
- 6
- 3
2
4
- 3
+ 1
- 3
- 9
4
12
`
D=^3
2
6
1
- 3
- 9
2
4
12
- 3
3
x+ 2 y- 3 z = 5
6 x- 9 y+ 12 z = 24
2 x- 3 y+ 4 z = 8
EXAMPLE 6
EA-^73 ,^223 , 7BF.
z=
Dz
D
=
- 21
- 3
y= =7.
Dy
D
=
- 22
- 3
=
22
3
x= ,
Dx
D
=
7
- 3
=-
7
3
,
Dy =- 22 Dz=-21.
= 7
=- 21 - 12 - 11 - 192 - 11142
= - 2 `
- 1
- 2
1
3
- 1
- 5
4
1
3
+ 1 - 12
- 5
4
- 1
- 2
`
Dx=^3
- 2
- 5
4
1
- 1
- 2
- 1
1
3
3
Dx
=- 3
= 11 - 12 - 1152 - 11 - 32
= (^1) `
- 1
- 2
1
3
(^) - (^1)
2
1
1
3
(^) + 1 - (^12)
2
1
- 1
- 2
(^) `
D=^3
1
2
1
1
- 1
- 2
- 1
1
3
3
720 APPENDIX A Determinants and Cramer’s Rule
NOW TRY
EXERCISE 6
Use Cramer’s rule, if possible,
to solve the system.
10 x+ 6 y+ 2 z= 3x- 2 y+ 3 z= 65 x+ 3 y+ z= 1NOW TRY ANSWERS
5.
6.Cramer’s rule does not apply
since D=0.
51 4, 0, - 126NOW TRY
EXERCISE 5
Use Cramer’s rule to solve the
system.
x- 3 y+ 4 z= 0- 2 x+ 5 y- 2 z=- 6
4 x+ 2 y+ z= 15Given system:x- 2 y+ 3 z= 42 x- y+ z=- 5x+ y- z=- 2