Correct Answer: Option C

Explanation

To evaluate \( 2M - 3N \) where 

\(M = \begin{pmatrix} 1 & 3 & 2 \\ 4 & 5 & -1 \\ -3 & 2 & 0 \end{pmatrix}\)

and  \(N = \begin{pmatrix} 1 & -2 & 3 \\ 4 & -1 & 5 \\ 2 & -3 & -1 \end{pmatrix}\)

 Calculate \( 2M \)

\(2M = 2 \times \begin{pmatrix} 1 & 3 & 2 \\ 4 & 5 & -1 \\ -3 & 2 & 0 \end{pmatrix} = \begin{pmatrix} 2 \cdot 1 & 2 \cdot 3 & 2 \cdot 2 \\ 2 \cdot 4 & 2 \cdot 5 & 2 \cdot -1 \\ 2 \cdot -3 & 2 \cdot 2 & 2 \cdot 0 \end{pmatrix} = \begin{pmatrix} 2 & 6 & 4 \\ 8 & 10 & -2 \\ -6 & 4 & 0 \end{pmatrix}\)

Calculate \( 3N \)

\(3N = 3 \times \begin{pmatrix} 1 & -2 & 3 \\ 4 & -1 & 5 \\ 2 & -3 & -1 \end{pmatrix} = \begin{pmatrix} 3 \cdot 1 & 3 \cdot -2 & 3 \cdot 3 \\ 3 \cdot 4 & 3 \cdot -1 & 3 \cdot 5 \\ 3 \cdot 2 & 3 \cdot -3 & 3 \cdot -1 \end{pmatrix} = \begin{pmatrix} 3 & -6 & 9 \\ 12 & -3 & 15 \\ 6 & -9 & -3 \end{pmatrix}\)

Calculate \( 2M - 3N \)

\(2M - 3N = \begin{pmatrix} 2 & 6 & 4 \\ 8 & 10 & -2 \\ -6 & 4 & 0 \end{pmatrix} - \begin{pmatrix} 3 & -6 & 9 \\ 12 & -3 & 15 \\ 6 & -9 & -3 \end{pmatrix}\)

Subtracting the matrices element-wise:

\(= \begin{pmatrix} 2 - 3 & 6 - (-6) & 4 - 9 \\ 8 - 12 & 10 - (-3) & -2 - 15 \\ -6 - 6 & 4 - (-9) & 0 - (-3) \end{pmatrix}\)

Calculating each element:

\(= \begin{pmatrix} -1 & 12 & -5 \\ -4 & 13 & -17 \\ -12 & 13 & 3 \end{pmatrix}\)

Final Result

Thus, the result of \( 2M - 3N \) is \(\(\begin{pmatrix} -1 & 12 & -5 \\ -4 & 13 & -17 \\ -12 & 13 & 3 \end{pmatrix}\)

There is an explanation video available below.

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