Chapter 7.4, Problem 104E

To determine

Whether the matrix is in reduced row echelon form, row echelon form or neither.

Answer to Problem 104E

When $b=0,c=0$ , the matrix is in reduced row echelon form.

When $b\ne 0,c=0$ , the matrix is in row echelon form.

When $b=0,c\ne 0$ , the matrix is neither in row echelon nor in reduced row echelon form.

When $b\ne 0,c\ne 0$ , the matrix is neither in row echelon nor in reduced row echelon form.

Explanation of Solution

Given information: The matrix is given by,

  $\left(\begin{array}{cc}1& b\\ c& 1\end{array}\right)$

Calculation:

When,

(a) $b=0,c=0$

The matrix can be written as,

  $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$

The matrix is in reduced row echelon form.

(b) $b\ne 0,c=0$

The matrix can be written as,

  $\left(\begin{array}{cc}1& b\\ 0& 1\end{array}\right)$

The matrix is in row echelon form.

(c) $b=0,c\ne 0$

The matrix can be written as,

  $\left(\begin{array}{cc}1& 0\\ c& 1\end{array}\right)$

Since the leading coefficient of second row is not 1 and is not right of the first non zero number in the row above, the matrix is neither in row echelon nor in reduced row echelon form.

(d)

  $b\ne 0,c\ne 0$

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