Q. 382 - Intermediate Algebra

Question

In the following exercise, perform the indicated operations on the augmented matrices.

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

ⓐ Interchange rows $2$ and $1$.

ⓑ Multiply row $1$ by $4$.

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

Step-by-Step Solution

Answer

Part a. The required augmented matrix is $\left[\begin{array}{lll}3& 2& 1\\ 4& -6& -3\end{array}\right]$

Part b. The required augmented matrix is $\left[\begin{array}{lll}16& -24& -12\\ 3& 2& 1\end{array}\right]$

Part c. The required augmented matrix is $\left[\begin{array}{lll}13& 0& 0\\ 3& 2& 1\end{array}\right]$

Part (a) Step 1. Finding the augmented matrix

In the given augmented matrix $\left[\begin{array}{lll}4& -6& -3\\ 3& 2& 1\end{array}\right]$, we interchange rows $2$ and $1$ and we get the new matrix as

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

Part (b) Step 1. Finding the augmented matrix

In the given augmented matrix $\left[\begin{array}{lll}4& -6& -3\\ 3& 2& 1\end{array}\right]$ we need to multiply $4$ in row $1$ and we get the new matrix as

$$\begin{gathered} \left[\begin{array}{lll}4& -6& -3\\ 3& 2& 1\end{array}\right]\stackrel{4R_1}{\to }\left[\begin{array}{lll}4·4& -6·4& -3·4\\ 3& 2& 1\end{array}\right] \\ \stackrel{4R_1}{\to }\left[\begin{array}{lll}16& -24& -12\\ 3& 2& 1\end{array}\right] \end{gathered}$$

Part (c) Step 1. Find the augmented matrix

For the given augmented matrix $\left[\begin{array}{lll}4& -6& -3\\ 3& 2& 1\end{array}\right]$, the objective is to multiply row $2$ by $3$ and add to row $1$. So by performing the operation we get

$$\begin{gathered} \left[\begin{array}{lll}4& -6& -3\\ 3& 2& 1\end{array}\right]\stackrel{3R_2+R_1}{\to }\left[\begin{array}{lll}3·3+4& 2·3+(-6)& 1·3+(-3)\\ 3& 2& 1\end{array}\right] \\ \stackrel{3R_2+R_1}{\to }\left[\begin{array}{lll}13& 0& 0\\ 3& 2& 1\end{array}\right] \end{gathered}$$