Questions: Perform the row operations on the matrix and write the resulting matrix. Replace R2 by 1/2 R1 + 1/2 R2 [ 2 0 6 -2 2 12 ] A. [ 2 0 6 0 0 9 ] B. [ 2 0 6 -1 1 6 ] C. [ 2 0 6 0 2 18 ] D. [ 2 0 6 0 1 9 ]

Perform the row operations on the matrix and write the resulting matrix.
Replace R2 by 1/2 R1 + 1/2 R2

[ 2 0 6 -2 2 12 ]

A. [ 2 0 6 0 0 9 ] B. [ 2 0 6 -1 1 6 ] C. [ 2 0 6 0 2 18 ] D. [ 2 0 6 0 1 9 ]

Transcript text: Perform the row operations on the matrix and write the resulting matrix. Replace $R_{2}$ by $\frac{1}{2} R_{1}+\frac{1}{2} R_{2}$ \[ \left[\begin{array}{rr|r} 2 & 0 & 6 \\ -2 & 2 & 12 \end{array}\right] \] $\qquad$ A. $\left[\begin{array}{ll|l}2 & 0 & 6 \\ 0 & 0 & 9\end{array}\right]$ B. $\left[\begin{array}{rr|r}2 & 0 & 6 \\ -1 & 1 & 6\end{array}\right]$ c. $\left[\begin{array}{rr|r}2 & 0 & 6 \\ 0 & 2 & 18\end{array}\right]$ D. $\left[\begin{array}{ll|l}2 & 0 & 6 \\ 0 & 1 & 9\end{array}\right]$

Solution

Solution Steps

To perform the specified row operation on the given matrix, we need to replace the second row $ R_2 $ with the average of the first row $ R_1 $ and the current second row $ R_2 $. This involves taking half of each element in $ R_1 $ and adding it to half of the corresponding element in $ R_2 $.

Step 1: Define the Original Matrix

The original matrix is given as: \[ \begin{bmatrix} 2 & 0 & 6 \\ -2 & 2 & 12 \end{bmatrix} \]

Step 2: Perform the Row Operation

We need to replace $ R_2 $ with $ \frac{1}{2} R_1 + \frac{1}{2} R_2 $. This can be calculated as follows: \[ R_2 = \frac{1}{2} \begin{bmatrix} 2 & 0 & 6 \end{bmatrix} + \frac{1}{2} \begin{bmatrix} -2 & 2 & 12 \end{bmatrix} \] Calculating each element:

First element: $ \frac{1}{2}(2) + \frac{1}{2}(-2) = 1 - 1 = 0 $
Second element: $ \frac{1}{2}(0) + \frac{1}{2}(2) = 0 + 1 = 1 $
Third element: $ \frac{1}{2}(6) + \frac{1}{2}(12) = 3 + 6 = 9 $

Thus, the new $ R_2 $ becomes: \[ R_2 = \begin{bmatrix} 0 & 1 & 9 \end{bmatrix} \]

Step 3: Write the Resulting Matrix

The resulting matrix after the row operation is: \[ \begin{bmatrix} 2 & 0 & 6 \\ 0 & 1 & 9 \end{bmatrix} \]