Question

question 31 points 2

\\(s = \

$$\begin{bmatrix} 5 & 2 \\\\ 2 & 1 \\end{bmatrix}$$

\\). does the matrix \\(s\\) have an inverse?

yes
no

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"
],
"new_concepts": [
"Matrix Inverse Existence"
],
"current_concepts": [
"Matrix Determinant",
"Matrix Inverse Existence"
]
}
</pre_analysis>

<reasoning>

Calculate the determinant of matrix S

Using the Matrix Determinant knowledge point
\[
\det(S) = (5)(1) - (2)(2) = 5 - 4 = 1
\]

Determine if the inverse exists

An inverse exists because the determinant is non-zero.
\[
\det(S) = 1
eq 0
\]
</reasoning>

<answer>
<mcq-correct>Yes</mcq-correct>
<mcq-option>No</mcq-option>
</answer>

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

<reasoning>

Calculate the determinant of matrix S

Using the Matrix Determinant knowledge point
\[
\det(S) = (5)(1) - (2)(2) = 5 - 4 = 1
\]

Determine if the inverse exists

An inverse exists because the determinant is non-zero.
\[
\det(S) = 1
eq 0
\]
</reasoning>

<answer>
<mcq-correct>Yes</mcq-correct>
<mcq-option>No</mcq-option>
</answer>

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