Chapter 7.4, Problem 102E

a.

To determine

The matrix $\left[\begin{array}{cc}1& b\\ c& 1\end{array}\right]$

is in row-echelon form, reduced row-echelon form, or neither.

Answer to Problem 102E

The matrix is in reduced row echelon form.

Explanation of Solution

Given information:

Condition given is $b=0$ , $c=0$

Given condition is $b=0$ , $c=0$

Therefore, the matrix according to given condition is

  $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

This matrix is in reduced row echelon form because each column that contains a leading 1 has zeroes everywhere else in that column.

b.

To determine

The matrix $\left[\begin{array}{cc}1& b\\ c& 1\end{array}\right]$

is in row-echelon form, reduced row-echelon form, or neither.

Answer to Problem 102E

The matrix is in row echelon form.

Explanation of Solution

Given information:

Condition given is $b\ne 0$ , $c=0$

Given condition is $b\ne 0$ , $c=0$

Let $b=2$

Therefore, the matrix according to given condition is

  $\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]$

This matrix is in row echelon form because the leading entry in each row is 1 .

c.

To determine

The matrix $\left[\begin{array}{cc}1& b\\ c& 1\end{array}\right]$

is in row-echelon form, reduced row-echelon form, or neither.

Answer to Problem 102E

The matrix is neither in row echelon form nor reduced row echelon form.

Explanation of Solution

Given information:

Condition given is $b=0$ , $c\ne 0$

Given condition is $b=0$ , $c\ne 0$

Let $c=2$

Therefore, the matrix according to given condition is

  $\left[\begin{array}{cc}1& 0\\ 2& 1\end{array}\right]$

This matrix is neither in row echelon form nor reduced row echelon form Because there are not any rows that consist entirely zeros also there is not any column that contains a leading 1 has zeroes in that column.

d.

To determine

The matrix $\left[\begin{array}{cc}1& b\\ c& 1\end{array}\right]$

is in row-echelon form, reduced row-echelon form, or neither.

Answer to Problem 102E

The matrix is neither in row echelon form nor in reduced row echelon form.

Explanation of Solution

Given information:

  $b\ne 0$ , $c\ne 0$

Given condition is $b\ne 0$ , $c\ne 0$

Let $b=3$ , $c=2$

Therefore, the matrix according to given condition is

  $\left[\begin{array}{cc}1& 3\\ 2& 1\end{array}\right]$

This matrix is neither in row echelon form nor in reduced row echelon form because there are not any rows that consist entirely zeros also there is not any column that contains a leading 1 has zeroes in that column.