Q 206 - Intermediate Algebra

Question

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

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

ⓐ Interchange rows $2$ and $3$ 

ⓑ Multiply row $1$ by $4$ 

ⓒ Multiply row $2$ by $-2$ and add to row $3$.

Step-by-Step Solution

Answer

Part (a) The required matrix is $\left[\begin{array}{rrrr}1& -3& -2& 4\\ 4& -2& -3& -1\\ 2& 2& -1& -3\end{array}\right]$

Part (b) The required matrix is $\left[\begin{array}{cccc}4& -12& -8& 16\\ 2& 2& -1& -3\\ 4& -2& -3& -1\end{array}\right]$

Part (c) The required matrix is $\left[\begin{array}{rrrr}1& -3& -2& 4\\ 2& 2& -1& -3\\ 0& -6& -1& 5\end{array}\right]$

Part (a) Step 1. Calculating Values

A matrix is given as 

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

Apply given operation as-

$\left[\begin{array}{rrrr}1& -3& -2& 4\\ 2& 2& -1& -3\\ 4& -2& -3& -1\end{array}\right]\stackrel{R_2\leftrightarrow R_3}{⟶}\left[\begin{array}{rrrr}1& -3& -2& 4\\ 4& -2& -3& -1\\ 2& 2& -1& -3\end{array}\right]$

Part (b) Step 1. Explanation

A matrix is given as

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

Apply given operation as-

$$\begin{gathered} \left[\begin{array}{rrrr}1& -3& -2& 4\\ 2& 2& -1& -3\\ 4& -2& -3& -1\end{array}\right]\stackrel{4R_1}{⟶}\left[\begin{array}{cccc}4\times 1& 4(-3)& 4(-2)& 4\times 4\\ 2& 2& -1& -3\\ 4& -2& -3& -1\end{array}\right] \\ \stackrel{4R_1}{⟶}\left[\begin{array}{cccc}4& -12& -8& 16\\ 2& 2& -1& -3\\ 4& -2& -3& -1\end{array}\right] \end{gathered}$$

Part (c) Step 1. Explanation

A matrix is given as

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

Apply given operation as-

$$\begin{gathered} \left[\begin{array}{rrrr}1& -3& -2& 4\\ 2& 2& -1& -3\\ 4& -2& -3& -1\end{array}\right]\stackrel{-2R_2+R_1}{⟶}\left[\begin{array}{llll}1& -3& -2& 4\\ 2& 2& -1& -3\\ (-2\times 2)+4& (-2\times 2)-2& (-2\times (-1\left)\right)-3& (-2\times (-3\left)\right)-1\end{array}\right] \\ \stackrel{-2R_2+R_1}{⟶}\left[\begin{array}{rrrr}1& -3& -2& 4\\ 2& 2& -1& -3\\ 0& -6& -1& 5\end{array}\right] \end{gathered}$$