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Exercises

Exercises

1.
Write augmented matrix for each system in Exercises 1.1--1.5
1.1

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

1.2

\begin{displaymath} \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. \end{displaymath}

1.3

\begin{displaymath} \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. \end{displaymath}

1.4

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

1.5

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

2.
Solve each system in Exercises 2.1--2.6 with the help of Gaussian Elimination Procedure.
2.1

\begin{displaymath} \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. \end{displaymath}

2.2

\begin{displaymath} \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. \end{displaymath}

2.3

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

2.4

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

2.5

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

2.6

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

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

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

3.2

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

3.3

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

3.4

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

3.5

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


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Sergey Nikitin 2004-01-28