To find the result of the expression [tex]\(3\left[\begin{array}{cc}5 & 6 \\ -9 & -2\end{array}\right] - 2\left[\begin{array}{cc}0 & 2 \\ -6 & -4\end{array}\right]\)[/tex], let's break it down step by step.
First, we'll perform the scalar multiplication separately on each matrix.
1. Multiply the first matrix by 3:
[tex]\[
\begin{array}{cc}
5 & 6 \\
-9 & -2 \\
\end{array}
\][/tex]
Multiplying each element by 3:
[tex]\[
3 \left[\begin{array}{cc}5 & 6 \\ -9 & -2\end{array}\right] = \left[\begin{array}{cc}3 \cdot 5 & 3 \cdot 6 \\ 3 \cdot (-9) & 3 \cdot (-2)\end{array}\right] = \left[\begin{array}{cc}15 & 18 \\ -27 & -6\end{array}\right]
\][/tex]
2. Multiply the second matrix by 2:
[tex]\[
\begin{array}{cc}
0 & 2 \\
-6 & -4 \\
\end{array}
\][/tex]
Multiplying each element by 2:
[tex]\[
2 \left[\begin{array}{cc}0 & 2 \\ -6 & -4\end{array}\right] = \left[\begin{array}{cc}2 \cdot 0 & 2 \cdot 2 \\ 2 \cdot (-6) & 2 \cdot (-4)\end{array}\right] = \left[\begin{array}{cc}0 & 4 \\ -12 & -8\end{array}\right]
\][/tex]
3. Subtract the second matrix from the first matrix:
[tex]\[
\left[\begin{array}{cc}15 & 18 \\ -27 & -6\end{array}\right] - \left[\begin{array}{cc}0 & 4 \\ -12 & -8\end{array}\right]
\][/tex]
Subtract corresponding elements:
[tex]\[
\left[\begin{array}{cc}15 - 0 & 18 - 4 \\ -27 - (-12) & -6 - (-8)\end{array}\right] = \left[\begin{array}{cc}15 & 14 \\ -15 & 2\end{array}\right]
\][/tex]
Therefore, the result of the expression is:
[tex]\[
\left[\begin{array}{cc}15 & 14 \\ -15 & 2\end{array}\right]
\][/tex]
So, the correct option is:
[tex]\[
\left[\begin{array}{cc}15 & 14 \\ -15 & 2\end{array}\right]
\][/tex]
