1. (a) Use Cramer's rule to solve for y in the SLE: 4х + y + 6 Зх + 7y 3 - + 2w 7x + 3y + 5z + 8w + y + 4 20

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Chapter2: Second-order Linear Odes
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1. (a) Use Cramer's rule to solve for y in the SLE:
4x + y
+
+
6.
Зх + 7у
+
3
-
y
+
+ 2w
4
7x + 3y + 5z + 8w
20
(b) Consider the following augmented matrix of an SLE in 3 variables.
а 0 ь 2
a 4 4
0а 2 b
a
Determine values of a and b such that the SLE has (i) a unique solution, (ii) a one-
parameter solution (one free variable), (iii) a two-parameter solution, (iv) is inconsistent.
2. Let
1 0 1 -1 2
1 0
6 1 9 -3 8
1
0 1 -1 2
A =
1 5
1 0
B=
2 15
о -2 1
1 1
(a) Show that A and B are not row-equivalent.
(b) Compute the values of rank(A) and rank(B).
(c) Compute the general solution for each of the SLES Ax =0 and Bx = 0.
(d) Find a solution of the SLE Bx = 0 that is not contained in the solution set of Ax = 0.
(e) Is the SLE Ax = b consistent for every be R°? Is the SLE Bx = b consistent for every
bER? In the affirmative case, justify your answer. In the negative case, find a vector
bER' such that the SLE is inconsistent.
Transcribed Image Text:1. (a) Use Cramer's rule to solve for y in the SLE: 4x + y + + 6. Зх + 7у + 3 - y + + 2w 4 7x + 3y + 5z + 8w 20 (b) Consider the following augmented matrix of an SLE in 3 variables. а 0 ь 2 a 4 4 0а 2 b a Determine values of a and b such that the SLE has (i) a unique solution, (ii) a one- parameter solution (one free variable), (iii) a two-parameter solution, (iv) is inconsistent. 2. Let 1 0 1 -1 2 1 0 6 1 9 -3 8 1 0 1 -1 2 A = 1 5 1 0 B= 2 15 о -2 1 1 1 (a) Show that A and B are not row-equivalent. (b) Compute the values of rank(A) and rank(B). (c) Compute the general solution for each of the SLES Ax =0 and Bx = 0. (d) Find a solution of the SLE Bx = 0 that is not contained in the solution set of Ax = 0. (e) Is the SLE Ax = b consistent for every be R°? Is the SLE Bx = b consistent for every bER? In the affirmative case, justify your answer. In the negative case, find a vector bER' such that the SLE is inconsistent.
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