Linear Algebra with Applications, 10th Edition

1.2

1. 1. Row echelon form: (a), (c), (d), (g), and (h); reduced row echelon form: (c), (d), and (g)

2. 2. 

   1. (a) inconsistent;

   2. (c) consistent, infinitely many solutions;

   3. (d) consistent $(4,5,2)$;

   4. (e) inconsistent;

   5. (f) consistent, $(5,3,2)$

3. 3. 

   1. (b) ∅;

   2. (c) $\{(2+3\alpha ,\alpha ,-2)|\alpha \text{ real}\}$;

   3. (d) $\{(5-2\alpha -\beta ,\alpha ,4-3\beta ,\beta )|\alpha ,\beta \text{ real}\}$;

   4. (e) $\{(3-5\alpha +2\beta ,\alpha ,\beta ,6)|\alpha ,\beta \text{ real}\}$;

   5. (f) $\{(\alpha ,2,-1)|\alpha \text{ real}\}$

4. 4. 

   1. (a) $x_1,x_2,x_3$ are lead variables.

   2. (c) $x_1,x_3$ are lead variables and x₂ is a free variable.

   3. (e) $x_1,x_4$ are lead variables and $x_2,x_3$ are free variables.

5. 5. 

   1. (a) $(5,1)$;

   2. (b) inconsistent;

   3. (c) $(0,0)$;

   4. (d) $\left\{({\displaystyle \frac{5-\alpha }{4}},{\displaystyle \frac{1+7\alpha }{8}},\alpha )|\alpha \text{ real}\right\}$;

   5. (e) $\{(8-2\alpha ,\alpha -5,\alpha )\}$;

   6. (f) inconsistent;

   7. (g) inconsistent;

   8. (h) inconsistent;

   9. (i) $(0,\frac{3}{2},1)$;

   10. (j) $\{(2-6\alpha ,4+\alpha ,3-\alpha ,\alpha )\}$;

   11. (k) $\{(\frac{15}{4}-\frac{5}{8}\alpha -\beta ,-\frac{1}{4}-\frac{1}{8}\alpha ,\alpha ,\beta )\}$;

6. 6. 

   1. (a) $(0,-1$