Chapter 7.3, Problem 101E

To determine

To describe: the error.

Answer to Problem 101E

So, the correct form of the system is,

  $\begin{array}{c}x-2y+3z=12.\\ y+3z=5.\\ z=2.\end{array}$

Now this system is in row-echelon form.

Explanation of Solution

Given information: The system,

  $\begin{array}{c}x-2y+3z=12.\\ y+3z=5.\\ 2z=4.\end{array}$

is in row-echelon form.

Formula used:

Row Echelon Form

A matrix is in row echelon form (ref) when it satisfies the following conditions.

• The first non-zero element in each row, called the leading entry, is 1.
• Each leading entry is in a column to the right of the leading entry in the previous row.
• Rows with all zero elements, if any, are below rows having a non-zero element

Each of the matrices shown below are examples of matrices in row echelon form.

  $\left[\begin{array}{cccc}1& 2& 3& 4\\ 0& 0& 1& 3\\ 0& 0& 0& 1\end{array}\right],\left[\begin{array}{cccc}1& 2& 3& 4\\ 0& 0& 1& 3\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right],\left[\begin{array}{cc}1& 2\\ 0& 1\\ 0& 0\end{array}\right]$

Calculation:

The given system is,

  $\begin{array}{c}x-2y+3z=12.\\ y+3z=5.\\ 2z=4.\end{array}$

The matrix representation of the given system is,

  $\left[\begin{array}{ccc}1& -2& 3\\ 0& 1& 3\\ 0& 0& 2\end{array}\right]\times \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}12\\ 5\\ 4\end{array}\right]$ .

So given system is not in row-echelon form.

To convert this given system in row-echelon form.

Reduce the coefficient of z from 2 to 1 by dividing third equation by 2.

  $\begin{array}{c}x-2y+3z=12.\\ y+3z=5.\\ z=2.\end{array}$

Now the matrix representation of the given system is,

  $\left[\begin{array}{ccc}1& -2& 3\\ 0& 1& 3\\ 0& 0& 1\end{array}\right]\times \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}12\\ 5\\ 2\end{array}\right]$ .

Now this system is in row-echelon form.

So, the error is in the coefficient of z in third equation.

So, the correct form of the system is,

  $\begin{array}{c}x-2y+3z=12.\\ y+3z=5.\\ z=2.\end{array}$

Now this system is in row-echelon form.