Chapter 7.4, Problem 40E

To determine

To calculate: To predict whether the given matrix is in row-echelon form or not. Further, if it is in row-echelon form, the objective is to predict if the matrix is in reduced row echelon form or not.

Answer to Problem 40E

The matrix is in row echelon form.

The matrix is not in reduced row echelon form.

Explanation of Solution

Given information: A matrix,

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

Formula used:

A matrix is said to be in row echelon form if it follows these three properties:

1. The leading coefficient, that is the first non-zero entry in the given matrix is 1.
2. In the second row, if there is any leading 1 it must be on the right of the one in the above row.
3. If there is any zero row, it must be below all the rows having non-zero entries.

A matrix is said to be in reduce row echelon form if it follows these four properties:

1. The leading coefficient, that is the first non-zero entry in the given matrix is 1.
2. In the second row, if there is any leading 1 it must be on the right of the one in the above row.
3. The leading entry in a row is the only non-zero entry in that column.
4. If there is any zero row, it must be below all the rows having non-zero entries.

Calculation: Consider the matrix,

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

Clearly, the leading entry in the matrix is 1.

The leading 1 in the second row is to the right of the 1 in the above matrix.

Hence, the matrix is in row-echelon form.

The matrix does not fulfil the third condition being in reduced row-echelon form.

Clearly, the leading entry in the matrix is 1.

The leading 1 in the second row is to the right of the 1 in the above matrix.

The leading entry in a row is not the only non-zero entry in that column.

Hence, the matrix is not in reduced row-echelon form.