To evaluate the expression [tex]\(\frac{1}{2}\left[\begin{array}{cc}-8 & 6 \\ 2 & 10\end{array}\right] + 3\left[\begin{array}{cc}-2 & 1 \\ 5 & -3\end{array}\right]\)[/tex], we will perform the operations step by step:
1. Scalar Multiplication of the First Matrix:
We need to multiply each entry in the matrix [tex]\(\left[\begin{array}{cc}-8 & 6 \\ 2 & 10\end{array}\right]\)[/tex] by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
\frac{1}{2}\left[\begin{array}{cc}-8 & 6 \\ 2 & 10\end{array}\right] = \left[\begin{array}{cc}\frac{1}{2} \times -8 & \frac{1}{2} \times 6 \\ \frac{1}{2} \times 2 & \frac{1}{2} \times 10\end{array}\right] = \left[\begin{array}{cc}-4 & 3 \\ 1 & 5\end{array}\right]
\][/tex]
2. Scalar Multiplication of the Second Matrix:
Next, we multiply each entry in the matrix [tex]\(\left[\begin{array}{cc}-2 & 1 \\ 5 & -3\end{array}\right]\)[/tex] by 3:
[tex]\[
3\left[\begin{array}{cc}-2 & 1 \\ 5 & -3\end{array}\right] = \left[\begin{array}{cc}3 \times -2 & 3 \times 1 \\ 3 \times 5 & 3 \times -3\end{array}\right] = \left[\begin{array}{cc}-6 & 3 \\ 15 & -9\end{array}\right]
\][/tex]
3. Adding the Two Resultant Matrices:
Finally, we add the resulting matrices from the previous two steps:
[tex]\[
\left[\begin{array}{cc}-4 & 3 \\ 1 & 5\end{array}\right] + \left[\begin{array}{cc}-6 & 3 \\ 15 & -9\end{array}\right]
\][/tex]
We add the corresponding entries:
[tex]\[
\left[\begin{array}{cc}-4 + -6 & 3 + 3 \\ 1 + 15 & 5 + -9\end{array}\right] = \left[\begin{array}{cc}-10 & 6 \\ 16 & -4\end{array}\right]
\][/tex]
Thus, the final result of evaluating [tex]\(\frac{1}{2}\left[\begin{array}{cc}-8 & 6 \\ 2 & 10\end{array}\right] + 3\left[\begin{array}{cc}-2 & 1 \\ 5 & -3\end{array}\right]\)[/tex] is:
[tex]\[
\left[\begin{array}{cc}-10 & 6 \\ 16 & -4\end{array}\right]
\][/tex]
