Chapter 8.1, Problem 48E

To determine

To write: the matrix in row echelon form.

Answer to Problem 48E

Row echelon form of the matrix is,

  $\left[\begin{array}{cccc}1& -3& 0& -7\\ 0& 1& 1& 2\\ 0& 0& 0& 0\end{array}\right]$

Explanation of Solution

Given information:

The given matrix is

  $\left[\begin{array}{cccc}1& -3& 0& -7\\ -3& 10& 1& 23\\ 4& -10& 2& -24\end{array}\right]$

Properties used:

A matrix is in row echelon form if it has the following properties:

1. Any rows consisting entirely of zeros occur at the bottom of the matrix.
2. For each row that does not consist entirely of zeros, the first nonzero entry is 1.
3. For two successive rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row.

Calculation:

Let us consider the following matrix

  $\left[\begin{array}{cccc}1& -3& 0& -7\\ -3& 10& 1& 23\\ 4& -10& 2& -24\end{array}\right]$

Therefore perform following steps to convert the matrix into row echelon form,

Perform $3R_1+R_2$ on row $R_2$

  $\left[\begin{array}{cccc}1& -3& 0& -7\\ 0& 1& 1& 2\\ 4& -10& 2& -24\end{array}\right]$

Perform $-4R_1+R_3$ on row $R_3$

  $\left[\begin{array}{cccc}1& -3& 0& -7\\ 0& 1& 1& 2\\ 0& 2& 2& 4\end{array}\right]$

Perform $\frac{1}{2}{R}_3$ on row $R_3$

  $\left[\begin{array}{cccc}1& -3& 0& -7\\ 0& 1& 1& 2\\ 0& 1& 1& 2\end{array}\right]$

Perform $-R_2+R_3$ on row $R_3$

  $\left[\begin{array}{cccc}1& -3& 0& -7\\ 0& 1& 1& 2\\ 0& 0& 0& 0\end{array}\right]$

Hence row echelon form of the matrix is,

  $\left[\begin{array}{cccc}1& -3& 0& -7\\ 0& 1& 1& 2\\ 0& 0& 0& 0\end{array}\right]$