Chapter 7.4, Problem 39E

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 39E

The matrix is in row echelon form.

The matrix is in reduced row echelon form.

Explanation of Solution

Given information: A matrix,

  $\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 1& 5\\ 0& 0& 0& 0\end{array}\right]$

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,

  $\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 1& 5\\ 0& 0& 0& 0\end{array}\right]$

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.

Also, all the non-zero rows are above the zero row.

Hence, the matrix is in row-echelon form.

The matrix fulfils all the condition for 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 the only non-zero entry in that column.

Also, all the non-zero rows are above the zero row.

Hence, the matrix is in reduced row-echelon form.