Q 207 - Intermediate Algebra

Question

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

$\left[\begin{array}{rrrr}6& -5& 2& 3\\ 2& 1& -4& 5\\ 3& -3& 1& -1\end{array}\right]$

ⓐ Interchange rows $2$ and $3$

ⓑ Multiply row $2$ by $5$ 

ⓒ Multiply row $3$ by $-2$ and add to row $1$. 

Step-by-Step Solution

Answer

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

Part (b) The required matrix is $\left[\begin{array}{cccl}6& -5& 2& 3\\ 10& 5& -20& 25\\ 3& -3& 1& 1\end{array}\right]$

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

Part (a) Step 1. Explanation

A matrix is given as $\left[\begin{array}{rrrr}6& -5& 2& 3\\ 2& 1& -4& 5\\ 3& -3& 1& 1\end{array}\right]$

Apply given operation as-

$\left[\begin{array}{rrrr}6& -5& 2& 3\\ 2& 1& -4& 5\\ 3& -3& 1& 1\end{array}\right]\stackrel{R_2\leftrightarrow R_1}{⟶}\left[\begin{array}{rrrr}6& -5& 2& 3\\ 3& -3& 1& 1\\ 2& 1& -4& 5\end{array}\right]$

Part (b) Step 1. Explanation

A matrix is given as $\left[\begin{array}{rrrr}6& -5& 2& 3\\ 2& 1& -4& 5\\ 3& -3& 1& 1\end{array}\right]$

Apply given operation as-

$$\begin{gathered} \left[\begin{array}{rrrr}6& -5& 2& 3\\ 2& 1& -4& 5\\ 3& -3& 1& 1\end{array}\right]\stackrel{S{R}_2}{⟶}\left[\begin{array}{llcl}6& -5& 2& 3\\ 5\times 2& 5\times 1& 5(-4)& 5\times 5\\ 3& -3& 1& 1\end{array}\right] \\ \stackrel{5R_2}{⟶}\left[\begin{array}{cccc}6& -5& 2& 3\\ 10& 5& -20& 25\\ 3& -3& 1& 1\end{array}\right] \end{gathered}$$

Part (c) Step 1. Explanation

A matrix is given as $\left[\begin{array}{rrrr}6& -5& 2& 3\\ 2& 1& -4& 5\\ 3& -3& 1& 1\end{array}\right]$

Apply given operation as-

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