Exercises

1.

Write augmented matrix for each system in Exercises $1.1$-$1.5.$

1.1

$$\left\{ \begin{array}{ccccc} 5\cdot x_1& +& 10 \cdot x_2 &=& 5 \\ x_1& +& 2\cdot x_2 &=& 1 \end{array}\right.$$

1.2

$$\left\{ \begin{array}{ccccccc} 5\cdot x_1 &+& 10 \cdot x_2 ... ...&=& 9 \\ 2\cdot x_1 &+& x_2 &+& x_3 &=& 1 \end{array}\right.$$

1.3

$$\left\{ \begin{array}{ccccccccc} 6\cdot x_1 &+& 12 \cdot x_... ...&+& 2\cdot x_2 &+& x_3 &+& 4\cdot x_4 &=& 4 \end{array}\right.$$

1.4

$$\left\{ \begin{array}{ccccccccccc} x_1 &+& x_2 &+& x_3 &+& ... ...& 27\cdot x_3 &+& 4\cdot x_4 &+& 8x_5 &=& 2 \end{array}\right.$$

1.5

$$\left\{ \begin{array}{ccccccccccccc} 5\cdot x_1 &+& 10 \cdo... ...& 15\cdot x_3 &+& x_4 &+& x_5 &+& x_6&=& 7 \end{array}\right.$$

2.

Solve each system in Exercises $2.1$-$2.6$ with the help of Gaussian Elimination Procedure.

2.1

$$\left\{ \begin{array}{ccccc} 2\cdot x_1& +& 3 \cdot x_2 &=& 1 \\ 3\cdot x_1& +& 4\cdot x_2 &=& 2 \end{array}\right.$$

2.2

$$\left\{ \begin{array}{ccccccc} x_1 &+& x_2 &+& x_3&=& 0 \\ ... ... x_1 &+& 4 \cdot x_2 &+& 9\cdot x_3 &=& 2 \end{array}\right.$$

2.3

$$\left\{ \begin{array}{ccccccccc} x_1 &+& 2 \cdot x_2 &+& 3\... ...=& 42\\ x_1 &+& x_2 &+& x_3 &+& x_4 &=& 4 \end{array}\right.$$

2.4

$$\left\{ \begin{array}{ccccccccccc} x_1 &+& x_2 &+& x_3 &+& ... ... x_3 &+& 9\cdot x_4 &+&16 \cdot x_5 &=& 17 \end{array}\right.$$

2.5

$$\left\{ \begin{array}{ccccccccccccc} 2\cdot x_1 &+& x_2 && ... ...& -1\\ && && && &&x_5 &+&2\cdot x_6&=& 1 \end{array}\right.$$

2.6

$$\left\{ \begin{array}{cccccccccccc} 5\cdot x_1& + &3x_2& && ... ...5&=&1 \\ && && && &3x_4&+&2\cdot x_5&=&1. \end{array}\right.$$

3.

In Exercises $3.1$-$3.5,$ given the augmented matrix of a system find out whether the system is consistent or inconsistent.

3.1

$$\left( \begin{array}{ccc} 1 & 2 & 1 \\ 1 & 4 & -1 \\ 1 ... ...t\vert \begin{array}{c} 4\\ 3\\ 2 \end{array}\right. \right)$$

3.2

$$\left( \begin{array}{cccc} 1 & 2 & 1 & 1 \\ 1 & 4 & -1 & ... ...t \begin{array}{c} 4\\ 3\\ 2\\ 1 \end{array}\right. \right)$$

3.3

$$\left( \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & ... ...{array}{c} 0\\ 1\\ 2\\ 3\\ 4\\ \end{array}\right. \right)$$

3.4

$$\left( \begin{array}{cccccc} 2 & 1 & 0 & 0 & 0 & 0 \\ 1 &... ...y}{c} 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ \end{array}\right. \right)$$

3.5

$$\left( \begin{array}{ccccccc} 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ ... ...\\ 1\\ 0\\ 1\\ 0\\ 1\\ 0\\ 1 \end{array}\right. \right)$$