Answered

Matrix [tex]\( C \)[/tex] is a transformation of matrix [tex]\( B \)[/tex], and matrix [tex]\( B \)[/tex] is a transformation of matrix [tex]\( A \)[/tex], as shown below.

[tex]\[
A = \begin{bmatrix}
2 & -2 & 4 & 6 \\
1 & 3 & 2 & 4 \\
2 & -1 & 4 & 6
\end{bmatrix}
\xrightarrow{\frac{1}{2} R_1 \rightarrow R_1} \text{Matrix } B
\xrightarrow{-R_1+R_2 \rightarrow R_2} \text{Matrix } C
\][/tex]

Which matrix represents matrix [tex]\( C \)[/tex]?

A. [tex]\(\begin{bmatrix} 1 & -1 & 2 & 3 \\ 1 & 3 & 2 & 4 \\ 2 & -1 & 4 & 6 \end{bmatrix}\)[/tex]

B. [tex]\(\begin{bmatrix} 2 & -2 & 4 & 6 \\ 0 & 4 & 0 & 1 \\ 2 & -1 & 4 & 6 \end{bmatrix}\)[/tex]



Answer :

GinnyAnswer
To determine which matrix represents matrix [tex]\( C \)[/tex], let's go through the transformations step-by-step.

1. Starting with Matrix [tex]\( A \)[/tex]:

[tex]\[ A = \begin{array}{ccc|c} 2 & -2 & 4 & 6 \\ 1 & 3 & 2 & 4 \\ 2 & -1 & 4 & 6 \end{array} \][/tex]

2. First Transformation: [tex]\(\frac{1}{2} R1 \rightarrow R1\)[/tex]
- Divide the first row by 2.

[tex]\[ B = \begin{array}{ccc|c} 1 & -1 & 2 & 3 \\ 1 & 3 & 2 & 4 \\ 2 & -1 & 4 & 6 \end{array} \][/tex]

3. Second Transformation: [tex]\(-R1 + R2 \rightarrow R2\)[/tex]
- Subtract the first row from the second row.

[tex]\[ R2 \rightarrow R2 - R1 = \begin{array}{ccc|c} 1 & -1 & 2 & 3 \\ (1 - 1) & (3 - (-1)) & (2 - 2) & (4 - 3) \\ 2 & -1 & 4 & 6 \end{array} = \begin{array}{ccc|c} 1 & -1 & 2 & 3 \\ 0 & 4 & 0 & 1 \\ 2 & -1 & 4 & 6 \end{array} \][/tex]

Thus, the resulting matrix [tex]\( C \)[/tex] is:

[tex]\[ C = \begin{array}{ccc|c} 1 & -1 & 2 & 3 \\ 0 & 4 & 0 & 1 \\ 2 & -1 & 4 & 6 \end{array} \][/tex]

The correct matrix representing matrix [tex]\( C \)[/tex] is:

[tex]\[ \left.\left\lvert\, \begin{array}{ccc|c} 1 & -1 & 2 & 3 \\ 0 & 4 & 0 & 1 \\ 2 & -1 & 4 & 6 \end{array}\right.\right] \][/tex]

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