Chapter 7.4, Problem 104E

a.

To determine

To determine: values of $a$ and $b$ for which the augmented matrix has exactly one solution.

Answer to Problem 104E

System has exactly one solution for all values of $b$ and $a$ such that $a\ne 0$ .

Explanation of Solution

Given information:

The augmented matrix is

  $\left[\begin{array}{cccc}1& 2& ⋮& -4\\ 0& a& ⋮& b\end{array}\right]$

The augmented matrix is

  $\left[\begin{array}{cccc}1& 2& ⋮& -4\\ 0& a& ⋮& b\end{array}\right]$

Consider $a=1$ and $b=2$

The matrix becomes

  $\left[\begin{array}{cccc}1& 2& ⋮& -4\\ 0& 1& ⋮& 2\end{array}\right]$

Which is consistent and has exactly one solution for $a=1$ and $b=2$

Hence in general this system has exactly one solution for all values of $b$ and $a$ such that $a\ne 0$ .

b.

To determine

To determine: values of $a$ and $b$ for which the augmented matrix has Infinitely many solutions .

Answer to Problem 104E

system has infinitely many solutions when $a=0$ and $b=0$

Explanation of Solution

Given information:

The augmented matrix is

  $\left[\begin{array}{cccc}1& 2& ⋮& -4\\ 0& a& ⋮& b\end{array}\right]$

The augmented matrix is

  $\left[\begin{array}{cccc}1& 2& ⋮& -4\\ 0& a& ⋮& b\end{array}\right]$

Consider $a=0$ and $b=0$

The matrix becomes

  $\left[\begin{array}{cccc}1& 2& ⋮& -4\\ 0& 0& ⋮& 0\end{array}\right]$

Which is consistent and contains free column thus the system above has a free variable and hence infinitely many solutions when $a=0$ and $b=0$ .

c.

To determine

To determine:All values of $a$ and $b$ for which the augmented matrix has no solution .

Answer to Problem 104E

System has no solution when $a=0$ and $b\ne 0$

Explanation of Solution

Given information:

The augmented matrix is

  $\left[\begin{array}{cccc}1& 2& ⋮& -4\\ 0& a& ⋮& b\end{array}\right]$

The augmented matrix is

  $\left[\begin{array}{cccc}1& 2& ⋮& -4\\ 0& a& ⋮& b\end{array}\right]$

Consider $a=0$ and $b=1$

The matrix becomes

  $\left[\begin{array}{cccc}1& 2& ⋮& -4\\ 0& 0& ⋮& 1\end{array}\right]$

Which is inconsistent ,thus for $a=0$ and $b=1$ system above has no solution. In general this system has no solution when $a=0$ and $b\ne 0$