Question

question 32 points 1

write the formula to find the inverse of a matrix \\(a\\).

\\(a^{-1} = \frac{1}{|a|} \text{adj } a\\)

\\(a^{-1} = 0\\)

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 Inverse Formula",
"Adjoint Matrix"
],
"new_concepts": [],
"current_concepts": [
"Matrix Inverse Formula",
"Adjoint Matrix"
]
}
</pre_analysis>

<reasoning>

State the definition of a matrix inverse

Using the Matrix Inverse Formula knowledge point
\[
A^{-1} A = A A^{-1} = I
\]

Express the inverse using the adjoint matrix

Using the Matrix Inverse Formula and Adjoint Matrix knowledge points
\[
A^{-1} = \frac{1}{|A|} \operatorname{adj} A
\]
</reasoning>

<answer>
<mcq-correct>(A) \(A^{-1} = \frac{1}{|A|} \operatorname{adj} A\)</mcq-correct>
<mcq-option>(B) \(A^{-1} = 0\)</mcq-option>
</answer>

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

<reasoning>

State the definition of a matrix inverse

Using the Matrix Inverse Formula knowledge point
\[
A^{-1} A = A A^{-1} = I
\]

Express the inverse using the adjoint matrix

Using the Matrix Inverse Formula and Adjoint Matrix knowledge points
\[
A^{-1} = \frac{1}{|A|} \operatorname{adj} A
\]
</reasoning>

<answer>
<mcq-correct>(A) \(A^{-1} = \frac{1}{|A|} \operatorname{adj} A\)</mcq-correct>
<mcq-option>(B) \(A^{-1} = 0\)</mcq-option>
</answer>

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