Question

find \\(8a\\) for \\(a = \

$$\begin{bmatrix} -3 & 4 & 5 \\\\ 1 & -2 & 3 \\\\ 7 & 8 & 9 \\\\ -5 & 6 & -7 \\end{bmatrix}$$

\\)

\\(\bigcirc \

$$\begin{bmatrix} -24 & 96 & 40 \\\\ 8 & -16 & 24 \\\\ 56 & 64 & 63 \\\\ -40 & 48 & -56 \\end{bmatrix}$$

\\)

\\(\bigcirc \

$$\begin{bmatrix} -24 & 32 & 40 \\\\ 8 & -16 & 24 \\\\ 56 & 64 & 63 \\\\ -40 & 48 & -56 \\end{bmatrix}$$

\\)

Explanation:

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<reasoning>

Identify the given matrix and scalar

Using the Scalar Multiplication knowledge point
\[
A =

$$\begin{bmatrix} -3 & 4 & 5 \\ 1 & -2 & 3 \\ 7 & 8 & 9 \\ -5 & 6 & -7 \end{bmatrix}$$

, \quad k = 8
\]

Multiply each element by the scalar

Using the Scalar Multiplication knowledge point
\[
8A =

$$\begin{bmatrix} 8(-3) & 8(4) & 8(5) \\ 8(1) & 8(-2) & 8(3) \\ 8(7) & 8(8) & 8(9) \\ 8(-5) & 8(6) & 8(-7) \end{bmatrix}$$

=

$$\begin{bmatrix} -24 & 32 & 40 \\ 8 & -16 & 24 \\ 56 & 64 & 72 \\ -40 & 48 & -56 \end{bmatrix}$$

\]

Compare with the given options

The options visible in the image are:
Option 1:
\[

$$\begin{bmatrix} -24 & 96 & 40 \\ 8 & -16 & 24 \\ 56 & 64 & 63 \\ -40 & 48 & -56 \end{bmatrix}$$

\]
Option 2:
\[

$$\begin{bmatrix} -24 & 32 & 40 \\ 8 & -16 & 24 \\ 56 & 64 & 63 \\ -40 & 48 & -56 \end{bmatrix}$$

\]
Note: Both options in the image contain a typo in the third row, third column element, showing \(63\) instead of \(72\) (since \(8 \times 9 = 72\)). Comparing the two options, Option 2 correctly has \(32\) in the first row, second column (since \(8 \times 4 = 32\)), whereas Option 1 has \(96\). Therefore, Option 2 is the intended correct choice.
</reasoning>

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

$$\begin{bmatrix} -24 & 96 & 40 \\ 8 & -16 & 24 \\ 56 & 64 & 63 \\ -40 & 48 & -56 \end{bmatrix}$$

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

$$\begin{bmatrix} -24 & 32 & 40 \\ 8 & -16 & 24 \\ 56 & 64 & 63 \\ -40 & 48 & -56 \end{bmatrix}$$

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

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"Mathematics",
"Algebra",
"Scalar Multiplication"
]
}
</post_analysis>

Answer:

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<reasoning>

Identify the given matrix and scalar

Using the Scalar Multiplication knowledge point
\[
A =

$$\begin{bmatrix} -3 & 4 & 5 \\ 1 & -2 & 3 \\ 7 & 8 & 9 \\ -5 & 6 & -7 \end{bmatrix}$$

, \quad k = 8
\]

Multiply each element by the scalar

Using the Scalar Multiplication knowledge point
\[
8A =

$$\begin{bmatrix} 8(-3) & 8(4) & 8(5) \\ 8(1) & 8(-2) & 8(3) \\ 8(7) & 8(8) & 8(9) \\ 8(-5) & 8(6) & 8(-7) \end{bmatrix}$$

=

$$\begin{bmatrix} -24 & 32 & 40 \\ 8 & -16 & 24 \\ 56 & 64 & 72 \\ -40 & 48 & -56 \end{bmatrix}$$

\]

Compare with the given options

The options visible in the image are:
Option 1:
\[

$$\begin{bmatrix} -24 & 96 & 40 \\ 8 & -16 & 24 \\ 56 & 64 & 63 \\ -40 & 48 & -56 \end{bmatrix}$$

\]
Option 2:
\[

$$\begin{bmatrix} -24 & 32 & 40 \\ 8 & -16 & 24 \\ 56 & 64 & 63 \\ -40 & 48 & -56 \end{bmatrix}$$

\]
Note: Both options in the image contain a typo in the third row, third column element, showing \(63\) instead of \(72\) (since \(8 \times 9 = 72\)). Comparing the two options, Option 2 correctly has \(32\) in the first row, second column (since \(8 \times 4 = 32\)), whereas Option 1 has \(96\). Therefore, Option 2 is the intended correct choice.
</reasoning>

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

$$\begin{bmatrix} -24 & 96 & 40 \\ 8 & -16 & 24 \\ 56 & 64 & 63 \\ -40 & 48 & -56 \end{bmatrix}$$

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

$$\begin{bmatrix} -24 & 32 & 40 \\ 8 & -16 & 24 \\ 56 & 64 & 63 \\ -40 & 48 & -56 \end{bmatrix}$$

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

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