Q 381 - Intermediate Algebra

Question

In the following exercise, 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 $2$.

ⓒ Multiply row $3$ by $-2$ and add

to row $2$.

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:

$\left[\begin{array}{cccc}2& -6& -4& 8\\ 2& 2& -1& -3\\ 4& -2& -3& -1\end{array}\right]$

Part (c). The resultant matrix is:

$\left[\begin{array}{llll}1& -3& -2& 4\\ -6& 6& 8& -8\\ -8& 4& 6& 2\end{array}\right]$

Part (a) Step 1. Interchange rows

Interchanging the row $2$ and $3$, we get:

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

Thus, the desired matrix is  $\left[\begin{array}{rrrr}1& -3& -2& 4\\ 4& -2& -3& -1\\ 2& 2& -1& -3\end{array}\right]$

Part (b) Step 1. Multiply row 1 by 2

Multiplying the row $1$ by  $2$ , we get:

$$\begin{gathered} \left[\begin{array}{rrrr}1& -3& -2& 4\\ 2& 2& -1& -3\\ 4& -2& -3& -1\end{array}\right]\stackrel{2R_1}{⟶}\left[\begin{array}{cccc}2·1& 2(-3)& 2(-2)& 2·4\\ 2& 2& -1& -3\\ 4& -2& -3& -1\end{array}\right] \\ =\left[\begin{array}{cccc}2& -6& -4& 8\\ 2& 2& -1& -3\\ 4& -2& -3& -1\end{array}\right] \end{gathered}$$

The resultant matrix is:

$\left[\begin{array}{cccc}2& -6& -4& 8\\ 2& 2& -1& -3\\ 4& -2& -3& -1\end{array}\right]$

Part (c). Step 1. Multiply the row 3 by − 2 and add to row 2 .

First, multiplying the row $3$ by $2$, we get:

$$\begin{gathered} \left[\begin{array}{rrrr}1& -3& -2& 4\\ 2& 2& -1& -3\\ 4& -2& -3& -1\end{array}\right]\stackrel{-2R_3}{⟶}\left[\begin{array}{rrrr}1& -3& -2& 4\\ 2& 2& -1& -3\\ 4(-2)& -2(-2)& -3(-2)& -1(-2)\end{array}\right] \\ =\left[\begin{array}{llll}1& -3& -2& 4\\ 2& 2& 2& -10\\ -8& 4& 6& 2\end{array}\right] \end{gathered}$$

Now add the row $3$ and $2$, we get:

$$\begin{gathered} \left[\begin{array}{llll}1& -3& -2& 4\\ 2& 2& 2& -10\\ -8& 4& 6& 2\end{array}\right]\stackrel{R_2\to R_2+R_3}{\to }\left[\begin{array}{llll}1& -3& -2& 4\\ 2+(-8)& 2+4& 2+6& -10+2\\ -8& 4& 6& 2\end{array}\right] \\ =\left[\begin{array}{llll}1& -3& -2& 4\\ -6& 6& 8& -8\\ -8& 4& 6& 2\end{array}\right] \end{gathered}$$

The resultant matrix is:

$\left[\begin{array}{llll}1& -3& -2& 4\\ -6& 6& 8& -8\\ -8& 4& 6& 2\end{array}\right]$