17.8 Exercises 497
Section 17.3
Evaluate:
Section 17.4
Use Cramer’s rule to solve the systems of equations:
- 3 x 1 − 12 y 1 − 12 z 1 = 10 16. w 1 + 12 x 1 + 13 y 1 + 1 z 1 = 15
x 1 + 1 y 1 − 1 z 1 = 102 w 1 + 1 x 1 + 1 y 1 + 1 z 1 = 13
2 x 1 + 12 y 1 + 1 z 1 = 10 w 1 + 12 x 1 + 1 y 1 = 14
x 1 + 1 y 1 + 12 z 1 = 10
- (i)Show that the following equations have no solution unlessk 1 = 13 , (ii)solve for this
value of k.
2 x 1 − 1 y 1 + 1 z 1 = 12
3 x 1 + 1 y 1 − 12 z 1 = 11
x 1 − 13 y 1 + 14 z 1 = 1 k
- (i)Find kfor which the following equations have a nontrivial solution, (ii)solve for this
value of k.
kx 1 + 15 y 1 + 13 z 1 = 10
5 x 1 + 1 y 1 − 1 z 1 = 10
kx 1 + 12 y 1 + 1 z 1 = 10
Find (i)the values of λfor which the following systems of equations have nontrivial solutions,
(ii)the solutions for these values of λ.
- 2 x 1 + 1 y 1 = 1 λx 20. 3 x 1 + 1 y 1 = 1 λx 21. x 1 + 12 y 1 − 13 z 1 = 1 λx
x 1 + 12 y 1 = 1 λyx 1 + 13 y 1 + 1 z 1 = 1 λy 2 x 1 + 14 y 1 − 16 z 1 = 1 λy
y 1 + 13 z 1 = 1 λz −x 1 − 12 y 1 + 13 z 1 = 1 λz
Section 17.5
Use the properties of determinants to show that:
25.Differentiate the following determinant with respect to x:
12 3
456
789
2345678xx
xxx
xxx
ab
cd
ab ab
cd cd
=
−+
−+
1
2
22 1
32 2
10 3
0
−
−−
=
36 3
21 5
12 1
0
−
−
=
−−
−
−
2617 5
0 3 22 17
00 4 12
00 0 6
11 0 0
32 2 3
21 1 2
5311
−
−
−
3400
1200
0031
0042
2013
3104
1 123
2210
−