Question

Use Row Operations on a Matrix

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

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

ⓐ Interchange rows $1$and $2$

ⓑ Multiply row $1$by $4$

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

Short answer

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

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

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

Step-by-step solution

Part (a) Step 1. Explanation

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

Apply given operation as-

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

Part (b) Step 1. Explanation

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

Apply given operation as-

$$\begin{gathered} \left[\begin{array}{rrr}4& -6& -3\\ 3& 2& 1\end{array}\right]\stackrel{4R_1}{⟶}\left[\begin{array}{lll}4\times 3& 4\times 2& 4\times 1\\ 3& 2& 1\end{array}\right] \\ \stackrel{4R_1}{⟶}\left[\begin{array}{lll}12& 8& 4\\ 3& 2& 1\end{array}\right] \end{gathered}$$

Part (c) Step 1. Explanation

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

Apply given operation as-

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