Q 209 - Intermediate Algebra

Question

Use Row Operations on a Matrix In the following exercises, perform the indicated operations on the augmented matrices .

Perform the needed row operations that will get the first entry in both row $2$ and row $3$ to be zero in the augmented matrix:

$\left[\begin{array}{rrrr}1& -2& 3& -4\\ 3& -1& -2& 5\\ 2& -3& -4& -1\end{array}\right]$

Step-by-Step Solution

Answer

The required matrix is $\left[\begin{array}{cccc}1& -2& 3& -4\\ 0& 5& -11& 17\\ 0& 1& -10& 7\end{array}\right]$

Step 1. Given Information

A matrix is given as $\left[\begin{array}{rrrr}1& -2& 3& -4\\ 3& -1& -2& 5\\ 2& -3& -4& -1\end{array}\right]$

Step 2. Explanation

A matrix is given as $\left[\begin{array}{rrrr}1& -2& 3& -4\\ 3& -1& -2& 5\\ 2& -3& -4& -1\end{array}\right]$

Apply given operation as-

$$\begin{gathered} \left[\begin{array}{rrrr}1& -2& 3& -4\\ 3& -1& -2& 5\\ 2& -3& -4& -1\end{array}\right]\underset{-2R_1+R_3}{\overset{-3R_3+R_2}{\to }}\left[\begin{array}{llll}1& -2& 3& -4\\ (-3\times 1)+3& (-3(-2\left)\right)-1& (-3\times 3)-2& (-3(-4\left)\right)+5\\ (-2\times 1)+2& (-2(-2\left)\right)-3& (-2\times 3)-4& (-2(-4\left)\right)-1\end{array}\right] \\ \stackrel{\stackrel{-3R_1+R_2}{-2R_1+R_3}}{\to }\left[\begin{array}{cccc}1& -2& 3& -4\\ 0& 5& -11& 17\\ 0& 1& -10& 7\end{array}\right] \end{gathered}$$