Question

question 5 points 3

find -n if n = \

$$\begin{bmatrix} 1 & 15 & 16 & -12 & -32 \\\\ 13 & -14 & 24 & -15 & 50 \\\\ 60 & -16 & 17 & -5 & 20 \\\\ 2 & 9 & -11 & 6 & -50 \\\\ 10 & -40 & -10 & -20 & 60 \\end{bmatrix}$$

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": [
"Scalar Multiplication"
],
"new_concepts": [
"Matrix Negation"
],
"current_concepts": [
"Scalar Multiplication",
"Matrix Negation"
]
}
</pre_analysis>

<reasoning>

Understand the matrix negation operation

Using the Scalar Multiplication knowledge point
\[
-N = -1 \cdot N
\]

Negate each element of the matrix

Using the Scalar Multiplication knowledge point
\[
N =

$$\begin{bmatrix} 1 & 15 & 16 & -12 & -32 \\ 13 & -14 & 24 & -15 & 50 \\ 60 & -16 & 17 & -5 & 20 \\ 2 & 9 & -11 & 6 & -50 \\ 10 & -40 & -10 & -20 & 60 \end{bmatrix}$$

\]
\[
-N =

$$\begin{bmatrix} -1 & -15 & -16 & 12 & 32 \\ -13 & 14 & -24 & 15 & -50 \\ -60 & 16 & -17 & 5 & -20 \\ -2 & -9 & 11 & -6 & 50 \\ -10 & 40 & 10 & 20 & -60 \end{bmatrix}$$

\]

Match with the given options

We compare our calculated matrix \(-N\) with the visible options:

• The first option has the last column as \([2, -2, -20, 50, -60]^T\), which is incorrect.
• The second option has the first three rows as:

\[

$$\begin{bmatrix} -1 & -15 & -16 & 12 & 32 \\ -13 & 14 & -24 & 15 & -50 \\ -60 & 16 & -17 & 5 & -20 \end{bmatrix}$$

\]
This matches our calculated matrix elements perfectly.
</reasoning>

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

$$\begin{bmatrix} -1 & -15 & -16 & 12 & 2 \\ -13 & 14 & -24 & 15 & -2 \\ -60 & 16 & -17 & 5 & -20 \\ -2 & -9 & 11 & -6 & 50 \\ -10 & 40 & 10 & 20 & -60 \end{bmatrix}$$

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

$$\begin{bmatrix} -1 & -15 & -16 & 12 & 32 \\ -13 & 14 & -24 & 15 & -50 \\ -60 & 16 & -17 & 5 & -20 \\ -2 & -9 & 11 & -6 & 50 \\ -10 & 40 & 10 & 20 & -60 \end{bmatrix}$$

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

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Scalar Multiplication"
]
}
</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": [
"Scalar Multiplication"
],
"new_concepts": [
"Matrix Negation"
],
"current_concepts": [
"Scalar Multiplication",
"Matrix Negation"
]
}
</pre_analysis>

<reasoning>

Understand the matrix negation operation

Using the Scalar Multiplication knowledge point
\[
-N = -1 \cdot N
\]

Negate each element of the matrix

Using the Scalar Multiplication knowledge point
\[
N =

$$\begin{bmatrix} 1 & 15 & 16 & -12 & -32 \\ 13 & -14 & 24 & -15 & 50 \\ 60 & -16 & 17 & -5 & 20 \\ 2 & 9 & -11 & 6 & -50 \\ 10 & -40 & -10 & -20 & 60 \end{bmatrix}$$

\]
\[
-N =

$$\begin{bmatrix} -1 & -15 & -16 & 12 & 32 \\ -13 & 14 & -24 & 15 & -50 \\ -60 & 16 & -17 & 5 & -20 \\ -2 & -9 & 11 & -6 & 50 \\ -10 & 40 & 10 & 20 & -60 \end{bmatrix}$$

\]

Match with the given options

We compare our calculated matrix \(-N\) with the visible options:

• The first option has the last column as \([2, -2, -20, 50, -60]^T\), which is incorrect.
• The second option has the first three rows as:

\[

$$\begin{bmatrix} -1 & -15 & -16 & 12 & 32 \\ -13 & 14 & -24 & 15 & -50 \\ -60 & 16 & -17 & 5 & -20 \end{bmatrix}$$

\]
This matches our calculated matrix elements perfectly.
</reasoning>

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

$$\begin{bmatrix} -1 & -15 & -16 & 12 & 2 \\ -13 & 14 & -24 & 15 & -2 \\ -60 & 16 & -17 & 5 & -20 \\ -2 & -9 & 11 & -6 & 50 \\ -10 & 40 & 10 & 20 & -60 \end{bmatrix}$$

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

$$\begin{bmatrix} -1 & -15 & -16 & 12 & 32 \\ -13 & 14 & -24 & 15 & -50 \\ -60 & 16 & -17 & 5 & -20 \\ -2 & -9 & 11 & -6 & 50 \\ -10 & 40 & 10 & 20 & -60 \end{bmatrix}$$

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

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