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Unformatted text preview: Math 136 Practice Problems # 9 213
1. Let A = 3 a −1. Assuming that A is invertible, ﬁnd A−1 by the cofactor method.
−2 1 4 4a + 1 −10 3 + 2a
14
−4 , so
Solution: We have cof(A) = −1
−1 − 3a 11 2a − 3 4a + 1 −1 −1 − 3a
11 (cof(A))T = −10 14
3 + 2 a −4 2 a − 3
Moreover, by deﬁnition of the determinant we have the determinant of A is the dot product of
the ith row of A with the ith column of (cof(A))T . Hence we get Hence, A−1 det A = 3(−1) + a(14) + (−1)(−4) = 1 + 14a 4a + 1 −1 −1 − 3a
1
1
11 .
= det A (cof(A))T = 1+14a −10 14
3 + 2a −4 2a − 3 2. Use Cramer’s Rule to solve the following systems.
a) 3x1 − x2 = 2
4x1 + 7x2 = 5 Solution: The coeﬃcient matrix is A = 3 −1
, so det A = 21 + 4 = 25. Hence,
47 1
25
1
x2 =
25 x1 = Thus, the solution is x = 19
2 −1
=
57
25
7
32
=
45
25 19/25
.
7/25 1 2 b) 2x1 + 3x2 + x3 = 1
x1 + x2 − x3 = −1
−2x1
+ 2 x3 = 1 231
Solution: The coeﬃcient matrix is A = 1 1 −1, so det A = 6. Hence,
−2 0 2
131
1
4
−1 1 −1 =
x1 =
6102
6
x2 = 2
1
1
1
−3
1 −1 −1 =
6 −2 1
6
2 x3 = 231
7
1
1 1 −1 =
6 −2 0 1
6 2/3
Hence, the solution is x = −1/2.
7/6 c) 2x1 + x2 = 1
3x1 + 7x2 = −2 Solution: The coeﬃcient matrix is A = 21
, so det A = 14 − 3 = 11. Hence,
37 1
11
1
x1 =
11
x1 = Thus, the solution is x = 9/11
.
−7/11 9
11
=
−2 7
11
7
21
=
3 −2
11 3 d) 5x1 + x2 − x3 = 4
9x1 + x2 − x3 = 1
x1 − x2 + 5 x3 = 2 5 1 −1
Solution: The coeﬃcient matrix is A = 9 1 −1, so det A = −16. Hence,
1 −1 5
4 1 −1
1
12
1 1 −1 =
x1 =
−16 2 −1 5
−16
x2 = 5 4 −1
1
−166
9 1 −1 =
−16 1 2 5
−16 x3 = 514
−42
1
9 1 1=
−16 1 −1 2
−16 −3/4
Thus, the solution is x = 83/7 .
21/8 x1
1
1a 2
0 1 b , x = x2 and b = −1. Assuming that A is invertible,
3. Let A =
2
x3
1 c −1
use Cramer’s Rule to ﬁnd the value of x2 in the solution of the equation Ax = b. 1a 2
Solution: We ﬁnd that det A = 0 1 b =
1 c −1
11
2
11
1
1
So, x2 = det A 0 −1 b = −3−b(c−a) 0 −1
1 2 −1
01 1
a
2
0
1
b = −3 − b(c − a).
0 c−a 3
2
3−b
b = −3−b(c−a) .
−3 ...
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This note was uploaded on 07/13/2011 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.
 Winter '08
 All
 Math

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