Question

express the equations \\(2x + y + z = 3\\), \\(5x - y + 4z = 2\\), and \\(4x - 3y + z = 1\\) as a matrix equation.

\\(\bigcirc\\) \\(\

$$\begin{bmatrix} 2 & 0 & 1 \\\\ 5 & -1 & 4 \\\\ 4 & -3 & 1 \\end{bmatrix}$$

\

$$\begin{bmatrix} x \\\\ y \\\\ z \\end{bmatrix}$$

= \

$$\begin{bmatrix} 3 \\\\ 2 \\\\ 1 \\end{bmatrix}$$

\\)

\\(\bigcirc\\) \\(\

$$\begin{bmatrix} 2 & 1 & 1 \\\\ 5 & -1 & 4 \\\\ 4 & -3 & 1 \\end{bmatrix}$$

\

$$\begin{bmatrix} x \\\\ y \\\\ z \\end{bmatrix}$$

= \

$$\begin{bmatrix} 3 \\\\ 2 \\\\ 1 \\end{bmatrix}$$

\\)

Explanation:

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<reasoning>

Identify the system of linear equations

Using the Coefficient Matrix knowledge point
\[

$$\begin{aligned} 2x + y + z &= 3 \\ 5x - y + 4z &= 2 \\ 4x - 3y + z &= 1 \end{aligned}$$

\]

Extract the coefficient matrix

Using the Coefficient Matrix knowledge point
\[
A =

$$\begin{bmatrix} 2 & 1 & 1 \\ 5 & -1 & 4 \\ 4 & -3 & 1 \end{bmatrix}$$

\]

Formulate the matrix equation

Using the Solving Matrix Equations knowledge point
\[

$$\begin{bmatrix} 2 & 1 & 1 \\ 5 & -1 & 4 \\ 4 & -3 & 1 \end{bmatrix}$$

$$\begin{bmatrix} x \\ y \\ z \end{bmatrix}$$

=

$$\begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(

$$\begin{bmatrix} 2 & 0 & 1 \\ 5 & -1 & 4 \\ 4 & -3 & 1 \end{bmatrix}$$

$$\begin{bmatrix} x \\ y \\ z \end{bmatrix}$$

=

$$\begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$$

\)</mcq-option>
<mcq-correct>(B) \(

$$\begin{bmatrix} 2 & 1 & 1 \\ 5 & -1 & 4 \\ 4 & -3 & 1 \end{bmatrix}$$

$$\begin{bmatrix} x \\ y \\ z \end{bmatrix}$$

=

$$\begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$$

\)</mcq-correct>
</answer>

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"question_type": "Multiple Choice",
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"Mathematics",
"Algebra",
"Coefficient Matrix"
]
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</post_analysis>

Answer:

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<reasoning>

Identify the system of linear equations

Using the Coefficient Matrix knowledge point
\[

$$\begin{aligned} 2x + y + z &= 3 \\ 5x - y + 4z &= 2 \\ 4x - 3y + z &= 1 \end{aligned}$$

\]

Extract the coefficient matrix

Using the Coefficient Matrix knowledge point
\[
A =

$$\begin{bmatrix} 2 & 1 & 1 \\ 5 & -1 & 4 \\ 4 & -3 & 1 \end{bmatrix}$$

\]

Formulate the matrix equation

Using the Solving Matrix Equations knowledge point
\[

$$\begin{bmatrix} 2 & 1 & 1 \\ 5 & -1 & 4 \\ 4 & -3 & 1 \end{bmatrix}$$

$$\begin{bmatrix} x \\ y \\ z \end{bmatrix}$$

=

$$\begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(

$$\begin{bmatrix} 2 & 0 & 1 \\ 5 & -1 & 4 \\ 4 & -3 & 1 \end{bmatrix}$$

$$\begin{bmatrix} x \\ y \\ z \end{bmatrix}$$

=

$$\begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$$

\)</mcq-option>
<mcq-correct>(B) \(

$$\begin{bmatrix} 2 & 1 & 1 \\ 5 & -1 & 4 \\ 4 & -3 & 1 \end{bmatrix}$$

$$\begin{bmatrix} x \\ y \\ z \end{bmatrix}$$

=

$$\begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$$

\)</mcq-correct>
</answer>

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"question_type": "Multiple Choice",
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"Algebra",
"Coefficient Matrix"
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