Question

question 37 points 3

find the determinant of the 3 by 3 matrix: \

$$\begin{bmatrix} 2 & 4 & 6 \\\\ 1 & 0 & 1 \\\\ 5 & 1 & 3 \\end{bmatrix}$$

.

14

12

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Matrix Determinant",
"Determinant Expansion"
],
"new_concepts": [],
"current_concepts": [
"Matrix Determinant",
"Determinant Expansion"
]
}
</pre_analysis>

<reasoning>

Define the matrix and target

Using the Matrix Determinant knowledge point
\[
A =

$$\begin{bmatrix} 2 & 4 & 6 \\ 1 & 0 & 1 \\ 5 & 1 & 3 \end{bmatrix}$$

\]
We need to find \(\det(A)\).

Expand along the second row

Using the Determinant Expansion knowledge point
\[

$$\begin{aligned} \det(A) &= -1 \cdot \det\begin{bmatrix} 4 & 6 \\ 1 & 3 \end{bmatrix} + 0 \cdot \det\begin{bmatrix} 2 & 6 \\ 5 & 3 \end{bmatrix} - 1 \cdot \det\begin{bmatrix} 2 & 4 \\ 5 & 1 \end{bmatrix} \end{aligned}$$

\]

Calculate the 2 by 2 determinants

Using the Matrix Determinant knowledge point
\[

$$\begin{aligned} \det\begin{bmatrix} 4 & 6 \\ 1 & 3 \end{bmatrix} &= 4(3) - 6(1) = 12 - 6 = 6\\ \det\begin{bmatrix} 2 & 4 \\ 5 & 1 \end{bmatrix} &= 2(1) - 4(5) = 2 - 20 = -18 \end{aligned}$$

\]

Compute final determinant value

Using the Matrix Determinant knowledge point
\[

$$\begin{aligned} \det(A) &= -1(6) + 0 - 1(-18)\\ &= -6 + 18\\ &= 12 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>14</mcq-option>
<mcq-correct>12</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Matrix Determinant"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Matrix Determinant",
"Determinant Expansion"
],
"new_concepts": [],
"current_concepts": [
"Matrix Determinant",
"Determinant Expansion"
]
}
</pre_analysis>

<reasoning>

Define the matrix and target

Using the Matrix Determinant knowledge point
\[
A =

$$\begin{bmatrix} 2 & 4 & 6 \\ 1 & 0 & 1 \\ 5 & 1 & 3 \end{bmatrix}$$

\]
We need to find \(\det(A)\).

Expand along the second row

Using the Determinant Expansion knowledge point
\[

$$\begin{aligned} \det(A) &= -1 \cdot \det\begin{bmatrix} 4 & 6 \\ 1 & 3 \end{bmatrix} + 0 \cdot \det\begin{bmatrix} 2 & 6 \\ 5 & 3 \end{bmatrix} - 1 \cdot \det\begin{bmatrix} 2 & 4 \\ 5 & 1 \end{bmatrix} \end{aligned}$$

\]

Calculate the 2 by 2 determinants

Using the Matrix Determinant knowledge point
\[

$$\begin{aligned} \det\begin{bmatrix} 4 & 6 \\ 1 & 3 \end{bmatrix} &= 4(3) - 6(1) = 12 - 6 = 6\\ \det\begin{bmatrix} 2 & 4 \\ 5 & 1 \end{bmatrix} &= 2(1) - 4(5) = 2 - 20 = -18 \end{aligned}$$

\]

Compute final determinant value

Using the Matrix Determinant knowledge point
\[

$$\begin{aligned} \det(A) &= -1(6) + 0 - 1(-18)\\ &= -6 + 18\\ &= 12 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>14</mcq-option>
<mcq-correct>12</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Matrix Determinant"
]
}
</post_analysis>