TEST 1 (LINEAR ALGEBRA)

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

Please show your calculations and stages in the process of solving each problem from the test.

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

System 1:

$$\begin{eqnarray*} &&5\cdot x_1+3x_2=1 \\ &&x_1+4\cdot x_2+x_3=1 \\ &&x_2+5\... ...3+x_4=1 \\ &&x_3+4\cdot x_4+x_5=1 \\ &&3x_4+5x_5=1. \\ && \end{eqnarray*}$$

System 2:

$$\begin{eqnarray*} x_1-3\cdot x_2+x_3 &=&2 \\ 3\cdot x_1+2\cdot x_2+x_3 &=&6 \\ 9\cdot x_1+4\cdot x_2+x_3 &=&13. \end{eqnarray*}$$

System 3:

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

System 4:

$$\begin{eqnarray*} x_1- x_2+x_3 +x_4+2x_5 &=&2 \\ x_1- x_2-2x_3-2x_4-4x_5 &=&1 \\ x_1+ x_2+4x_3+4x_4+10x_5 &=&1 \\ x_1- x_2+8x_3-8x_4 &=&1 \end{eqnarray*}$$

System 5:

$$\begin{eqnarray*} x_1+ x_2+x_3 +x_4+2x_5 &=&0 \\ x_1+ x_2-2x_3-2x_4-2x_5 &=&0 \\ x_1+ x_2+8x_3-8x_4+2x_5 &=&0 \end{eqnarray*}$$

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

a) $A=\left( \begin{array}{cc} 2 & 4 \\ 1 & 8 \\ 1 & 6 \end{array} \right)... ... \begin{array}{cccc} 0 & 1 & 5 & 1 \\ 1 & 5 & 2 & 3 \end{array} \right) .$

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

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

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

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

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

5.Calculate the following determinants.

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

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

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