To find [tex]\( 2A + B \)[/tex] where [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are given matrices, we will follow these steps:
1. Matrix [tex]\( A \)[/tex]:
[tex]\[
A = \left[\begin{array}{cc}4 & 2 \\ -1 & 0\end{array}\right]
\][/tex]
2. Matrix [tex]\( B \)[/tex]:
[tex]\[
B = \left[\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right]
\][/tex]
3. Calculate [tex]\( 2A \)[/tex]:
To find [tex]\( 2A \)[/tex], multiply each element of matrix [tex]\( A \)[/tex] by 2:
[tex]\[
2A = 2 \cdot \left[\begin{array}{cc}4 & 2 \\ -1 & 0\end{array}\right] = \left[\begin{array}{cc}2 \cdot 4 & 2 \cdot 2 \\ 2 \cdot (-1) & 2 \cdot 0\end{array}\right] = \left[\begin{array}{cc}8 & 4 \\ -2 & 0\end{array}\right]
\][/tex]
4. Add [tex]\( 2A \)[/tex] and [tex]\( B \)[/tex]:
Now add matrix [tex]\( B \)[/tex] to [tex]\( 2A \)[/tex]:
[tex]\[
2A + B = \left[\begin{array}{cc}8 & 4 \\ -2 & 0\end{array}\right] + \left[\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right] = \left[\begin{array}{cc}8 + 1 & 4 + 2 \\ -2 + 3 & 0 + 4\end{array}\right]
\][/tex]
5. Compute each element of the resulting matrix:
[tex]\[
2A + B = \left[\begin{array}{cc}9 & 6 \\ 1 & 4\end{array}\right]
\][/tex]
So, the resulting matrix [tex]\( 2A + B \)[/tex] is:
[tex]\[
\left[\begin{array}{cc}9 & 6 \\ 1 & 4\end{array}\right]
\][/tex]
