Q 204 - Intermediate Algebra

Question

Use Row Operations on a Matrix

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

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

a Interchange rows $1$ and $2$

b Multiply row $2$ by $3$

c Multiply row $2$ by $-2$ and add

to row $1$.

Step-by-Step Solution

Answer

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

Part (b) The required matrix is $\left[\begin{array}{lll}6& -4& 3\\ 9& -6& 3\end{array}\right]$

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

Part (a) Step 1. Explanation

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

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

Part (b) Step 1. Explanation

A matrix is given as

Apply given operation as-

$\left[\begin{array}{lll}6& -4& 3\\ 3& -2& 1\end{array}\right]\stackrel{3R_2}{⟶}\left[\begin{array}{lll}6& -4& 3\\ 9& -6& 3\end{array}\right]$

Part (c) Step 1. Explanation

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

Apply given operation as-

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