Chapter 7.3, Problem 102E

Solution

To determine: The statement true or false that the reduced row echelon from of the augmented matrix of a system of three linear equations in three variables must be of the form.

The given statement is true.

Given Information:

The matrix is defined as,

  $\left[\begin{array}{cccc}1& 0& 0& a\\ 0& 1& 0& b\\ 0& 0& 1& c\end{array}\right]$

Explanation:

Consider the given information,

  $\left[\begin{array}{cccc}1& 0& 0& a\\ 0& 1& 0& b\\ 0& 0& 1& c\end{array}\right]$

Reduced row echelon form consists of the first row starting off with a 1 followed by 0's until the last column which contains a real number, the second row starts off with a 0 in the first column and a 1 in the second column followed by 0's until the last column, this continues on depending on the number of rows and columns.

In the problem it is set up this way replacing the last column with variables which stand for a real number which makes the equation true.

Therefore, the given statement is true.