Chapter 7.4, Problem 40E

To determine

Whether the matrix is in row echelon form or reduced row echelon form or not.

Answer to Problem 40E

The given matrix is in row echelon form but not in reduced row echelon form.

Explanation of Solution

Given information:

Given matrix

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

Calculation:

Consider the following matrices,

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

We know that a matrix is in row echelon form if it has the following properties:

1. Any rows consisting entirely of zeros occur at the bottom of the matrix.
2. For each row that does not consist entirely of zeros, the first nonzero entry is 1.
3. For two successive rows, the leading 1 in the higher row is further to the left than the leading 1 in the lower row.

And a matrix in row-echelon form is in reduced row echelon form if every column that has a leading 1 has zeros in every position above and below its leading 1.

Hence, according to the above rule, the above matrix is in row echelon form but not in reduced row echelon form.