TEST 1 (MAT 343)

Instr. S. Nikitin

ASU ID:..................................NAME:..................................

1. Find all of the solutions, if any exist, for the following systems.

System 1:

$$\begin{eqnarray*} &&4\cdot x_1+x_2=1 \\ &&x_1+4\cdot x_2+x_3=1 \\ &&x_2+4\c... ...=-1 \\ &&x_4+4\cdot x_5+x_6=1 \\ &&x_5+4\cdot x_6=1 \\ && \end{eqnarray*}$$

System 2:

$$\begin{eqnarray*} &&-2\cdot x_1+x_2=1 \\ &&x_1-2\cdot x_2+x_3=1 \\ &&x_2-x_... ...-1 \\ &&x_3-2\cdot x_4+x_5=1 \\ &&x_4-2\cdot x_5=1. \\ && \end{eqnarray*}$$

System 3:

$$\begin{eqnarray*} x_1+4\cdot x_2+5\cdot x_3 &=&1 \\ 2\cdot x_1+4\cdot x_2+x_3 &=&1 \\ 4\cdot x_1+4\cdot x_2+x_3 &=&2. \end{eqnarray*}$$

System 4:

$$\begin{eqnarray*} x_1+ x_2-x_3 &=&2 \\ -2\cdot x_1+2\cdot x_2+x_3 &=&1 \\ 3\cdot x_1+2\cdot x_2+x_3 &=&10. \end{eqnarray*}$$

System 5:

$$\begin{eqnarray*} 2x_1+ x_2+x_3 +x_4 &=&2 \\ 2x_1- x_2+2x_3-2x_4 &=&0 \\ 2x_1+ x_2+4x_3+4x_4 &=&1 \\ 2x_1- x_2+8x_3-8x_4 &=&1 \end{eqnarray*}$$

System 6:

$$\begin{eqnarray*} x_1+ 3\cdot x_2+x_3 +x_4 &=&2 \\ x_1- 3\cdot x_2+x_3-x_4 &=... ... 3\cdot x_2+4x_3+4x_4 &=&0 \\ x_1- 3\cdot x_2+7x_3-8x_4 &=&1 \end{eqnarray*}$$

2.Given matrices $A,\;\;B$ calculate $A\cdot B.$

a) $A=\left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \\ 1 & 0 \end{array} \right... ... \begin{array}{cccc} 1 & 0 & -1 & 0 \\ 1 & 1 & 0 & 1 \end{array} \right) .$

b) $A=\left( \begin{array}{cccc} -1 & 1 & 0 & 0 \\ 2 & -1 & 1 & 0 \\ 0 & 0 &... ... -1 \end{array} \right) . \begin{array}{cc} & \\ & \\ & \end{array}$

c) $A=\left( \begin{array}{cccc} 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0... ... 1 & 0 \\ 1 & -1 & 0 \\ 0 & 1 & 0 \\ 1 & -1 & 1 \end{array} \right) .$

3.Given a matrix $A$ find $A^{-1}.$

a) $A=\left( \begin{array}{cccc} 2 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 2 \end{array} \right) .$ b) $A=\left( \begin{array}{ccccc} 3 & 1 & 0 & 0 & 0 \\ 1 & -1 & 1 & 0 & 0 \\ ... ...& 0 \\ 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 1 & -1 \end{array} \right) .$

4.Use Cramer's Rule to find the solution of the systems

$$\begin{array}{cccccc} 3\cdot x_1+ & x_2+ & x_3+ &x_4& = & ... ...+ & x_3+ &3x_4& = & 1 \\ & & & & \\ & & & & \end{array}$$

$$\begin{array}{cccccc} 4x_1+ & x_2+ & x_3+ &4x_4& = & 1 \\ ... ...+ & x_3+ &4x_4& = & 1 \\ & & & & \\ & & & & \end{array}$$

5.Calculate the following determinants.

$$\det \left( \begin{array}{cccc} 3 & 1 & 0 & 1 \\ 1 & 3... ... \\ 0 & 0 & 4 & 1 \\ 1 & -2& 1 & 4 \end{array} \right),$$

$$\det \left( \begin{array}{ccccc} 3 & 1 & 0 & 0 & 1 \\ ... ... 8 & 0 & 5 & 1 \\ 1 & 16 & 0 & 0 & 5 \end{array} \right)$$

6. Find all solutions (if any exist) for the following linear systems.

$$\begin{eqnarray*} -3\cdot x_1+ x_2+x_3 +x_4 &=&1 \\ 2x_1- 3x_2+x_3 &=&0 \\ -x_1- x_2+3x_3-x_4 &=&1 \\ x_1- 2x_2+x_3 &=&0 \end{eqnarray*}$$

$$\begin{eqnarray*} x_1+ 2x_2+3x_3 +4x_4 &=&0 \\ 4x_1- 4x_2+2x_3 &=&0 \\ x_1+ 2x_2+4x_3+4x_4 &=&0 \\ x_1- x_2+8x_3 &=&0 \end{eqnarray*}$$