Chapter 8.1, Problem 29E

To determine

To identify: the elementary row operations performed to obtain the new row equivalent matrix.

Answer to Problem 29E

The elementary row operations performed to obtain the new row equivalent matrix are

  $R_1\to R_2$ and $4R_1+R_3\to R_3$

Explanation of Solution

Given information:

The given matrix is

  $\left[\begin{array}{cccc}0& -1& -5& 5\\ -1& 3& -7& 6\\ 4& -5& 1& 3\end{array}\right]$ and $\left[\begin{array}{cccc}0& -1& -5& 5\\ -1& 3& -7& 6\\ 4& -5& 1& 3\end{array}\right]$

Calculation:

Let us consider the following row-equivalent matrices

Original Matrix $\left[\begin{array}{cccc}0& -1& -5& 5\\ -1& 3& -7& 6\\ 4& -5& 1& 3\end{array}\right]$

New row equivalent Matrix $\left[\begin{array}{cccc}0& -1& -5& 5\\ -1& 3& -7& 6\\ 4& -5& 1& 3\end{array}\right]$

Two matrices are called row equivalent if one of the matrices can be obtained from the other by a sequence of elementary row operations.

Solve the above matrix further by using row operations as

  $\begin{array}{l}{R}_1\to R_2\left[\begin{array}{cccc}-1& 3& -7& 6\\ 0& -1& -5& 5\\ 4& -5& 1& 3\end{array}\right]\\ 4R_1+R_3\to R_3\left[\begin{array}{cccc}-& 3& 7& 6\\ 0& -1& -5& 5\\ 0& 7& -27& 27\end{array}\right]\end{array}$

Hence, the elementary row operations performed to obtain the new row equivalent matrix are

  $R_1\to R_2$ and $4R_1+R_3\to R_3$

And in words, we say “ interchange Row 1 and Row 2” and “Add 4 times new Row 1 to Row 3”.