1.2

1. Row echelon form: (a), (c), (d), (g), and (h); reduced row echelon form: (c), (d), and (g)
2.
1. (a) inconsistent;
2. (c) consistent, infinitely many solutions;
3. (d) consistent $(4, 5, 2)$ $(4, 5, 2)$ ;
4. (e) inconsistent;
5. (f) consistent, $(5, 3, 2)$ $(5, 3, 2)$
3.
1. (b) ∅;
2. (c) ${(2 + 3 α, α, - 2) | α real}$ ${(2 + 3 α, α, - 2) | α real}$ ;
3. (d) ${(5 - 2 α - β, α, 4 - 3 β, β) | α, β real}$ ${(5 - 2 α - β, α, 4 - 3 β, β) | α, β real}$ ;
4. (e) ${(3 - 5 α + 2 β, α, β, 6) | α, β real}$ ${(3 - 5 α + 2 β, α, β, 6) | α, β real}$ ;
5. (f) ${(α, 2, - 1) | α real}$ ${(α, 2, - 1) | α real}$
4.
1. (a) $x_{1}, x_{2}, x_{3}$ $x_{1}, x_{2}, x_{3}$ are lead variables.
2. (c) $x_{1}, x_{3}$ $x_{1}, x_{3}$ are lead variables and x₂ is a free variable.
3. (e) $x_{1}, x_{4}$ $x_{1}, x_{4}$ are lead variables and $x_{2}, x_{3}$ $x_{2}, x_{3}$ are free variables.
5.
1. (a) $(5, 1)$ $(5, 1)$ ;
2. (b) inconsistent;
3. (c) $(0, 0)$ $(0, 0)$ ;
4. (d) ${(\frac{5 - α}{4}, \frac{1 + 7 α}{8}, α) | α real}$ ${(\frac{5 - α}{4}, \frac{1 + 7 α}{8}, α) | α real}$ ;
5. (e) ${(8 - 2 α, α - 5, α)}$ ${(8 - 2 α, α - 5, α)}$ ;
6. (f) inconsistent;
7. (g) inconsistent;
8. (h) inconsistent;
9. (i) $(0, \frac{3}{2}, 1)$ $(0, \frac{3}{2}, 1)$ ;
10. (j) ${(2 - 6 α, 4 + α, 3 - α, α)}$ ${(2 - 6 α, 4 + α, 3 - α, α)}$ ;
11. (k) ${(\frac{15}{4} - \frac{5}{8} α - β, - \frac{1}{4} - \frac{1}{8} α, α, β)}$ ${(\frac{15}{4} - \frac{5}{8} α - β, - \frac{1}{4} - \frac{1}{8} α, α, β)}$ ;
6.
1. (a) $(0, - 1)$ $(0, - 1)$ ;
2. (b) ${(\frac{3}{4} - \frac{5}{8} α, - \frac{1}{4} - \frac{1}{8} α, α, 3) | α is real}$ ${(\frac{3}{4} - \frac{5}{8} α, - \frac{1}{4} - \frac{1}{8} α, α, 3) | α is real}$ ;
3. (d) ${α (- \frac{4}{3}, 0, \frac{1}{3}, 1)}$ ${α (- \frac{4}{3}, 0, \frac{1}{3}, 1)}$
7. Hint: The hint for Exercise11 in the previous section also applies to this exercise.
8. $a \neq - 2$ $a \neq - 2$
9.
1. (b) $β = 2$ $β = 2$
10.
1. (a) $a = 5, b = 4$ $a = 5, b = 4$ ;
2. (b) $a = 5, b \neq 4$ $a = 5, b \neq 4$
11.
1. (a) $(- 2, 2)$ $(- 2, 2)$ ;
2. (b) $(- 7, 4)$ $(- 7, 4)$
12.
1. (a) $(- 3, 2, 1)$ $(- 3, 2, 1)$ ;
2. (b) $(2, - 2, 1)$ $(2, - 2, 1)$
13. Hint: Is a homogeneous system necessarily consistent? Could the row echelon form for the coefficient matrix possibly involve free variables?
14. Hint: Will the row echelon form of the coefficient matrix involve free variables?
15. $x_{1} = 280, x_{2} = 230, x_{3} = 350, x_{4} = 590$ $x_{1} = 280, x_{2} = 230, x_{3} = 350, x_{4} = 590$
17. Hint: Plug in $x_{1} = α c_{1}$ $x_{1} = α c_{1}$ and $x_{2} = α c_{2}$ $x_{2} = α c_{2}$ into the system to see if they work.
19. $x_{1} = 2, x_{2} = 3, x_{3} = 12, x_{4} = 6$ $x_{1} = 2, x_{2} = 3, x_{3} = 12, x_{4} = 6$
20. 6 moles $N_{2}$ $N_{2}$ , 18 moles $H_{2}$ $H_{2}$ , 21 moles $O_{2}$ $O_{2}$
21. All three should be equal, i.e., $x_{1} = x_{2} = x_{3}$ $x_{1} = x_{2} = x_{3}$ .
22.
1. (a) $(5, 3, - 2)$ $(5, 3, - 2)$ ;
2. (b) $(2, 4, 2)$ $(2, 4, 2)$ ;
3. (c) $(2, 0, - 2, - 2, 0, 2)$ $(2, 0, - 2, - 2, 0, 2)$

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Table of Contents for
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