Given the matrices $A$ and $B$ shown below, find $B+\frac{1}{2} A$.
\[
A=\left[\begin{array}{cc}
0 & 6 \\
0 & -2 \\
-10 & 12
\end{array}\right] \quad B=\left[\begin{array}{cc}
-5 & 9 \\
8 & 9 \\
-7 & -2
\end{array}\right]
\]

Step 1 ：Given the matrices $A$ and $B$ as follows:

Step 2 ：$A=\left[\begin{array}{cc} 0 & 6 \\ 0 & -2 \\ -10 & 12 \end{array}\right]$ and $B=\left[\begin{array}{cc} -5 & 9 \\ 8 & 9 \\ -7 & -2 \end{array}\right]$

Step 3 ：We are asked to find the sum of matrix $B$ and half of matrix $A$. To do this, we first need to find half of matrix $A$, and then add it to matrix $B$.

Step 4 ：We can find half of matrix $A$ by multiplying each element of matrix $A$ by 0.5. This gives us the following matrix:

Step 5 ：$\frac{1}{2}A=\left[\begin{array}{cc} 0 & 3 \\ 0 & -1 \\ -5 & 6 \end{array}\right]$

Step 6 ：We then add the corresponding elements of this matrix to matrix $B$ to get the final result:

Step 7 ：$B+\frac{1}{2}A=\left[\begin{array}{cc} -5 & 12 \\ 8 & 8 \\ -12 & 4 \end{array}\right]$

Step 8 ：So, the result of $B+\frac{1}{2}A$ is $\boxed{\begin{array}{cc} -5 & 12 \\ 8 & 8 \\ -12 & 4 \end{array}}$