To find the elements [tex]\( C_{12}, C_{31}, \)[/tex] and [tex]\( C_{22} \)[/tex] of the resulting matrix [tex]\( C \)[/tex], we should first add the corresponding elements of the given matrices.
Given matrices:
[tex]\[
C_1 = \begin{bmatrix}
16 & 9 \\
-3 & 0 \\
4 & -10
\end{bmatrix}
\][/tex]
[tex]\[
C_2 = \begin{bmatrix}
-0.5 & 0 \\
5 & 8 \\
-3 & 14
\end{bmatrix}
\][/tex]
The addition of these matrices is done element-wise to get matrix [tex]\( C \)[/tex]:
[tex]\[
C = C_1 + C_2 = \begin{bmatrix}
16 & 9 \\
-3 & 0 \\
4 & -10
\end{bmatrix} + \begin{bmatrix}
-0.5 & 0 \\
5 & 8 \\
-3 & 14
\end{bmatrix}
\][/tex]
[tex]\[
C = \begin{bmatrix}
16 + (-0.5) & 9 + 0 \\
-3 + 5 & 0 + 8 \\
4 + (-3) & -10 + 14
\end{bmatrix}
\][/tex]
Let's compute each element:
[tex]\[
\begin{bmatrix}
16 + (-0.5) & 9 + 0 \\
-3 + 5 & 0 + 8 \\
4 + (-3) & -10 + 14
\end{bmatrix} = \begin{bmatrix}
15.5 & 9 \\
2 & 8 \\
1 & 4
\end{bmatrix}
\][/tex]
Now, we can extract the required elements:
[tex]\[
C_{12} = \text{Element in the first row, second column} = 9
\][/tex]
[tex]\[
C_{31} = \text{Element in the third row, first column} = 1
\][/tex]
[tex]\[
C_{22} = \text{Element in the second row, second column} = 8
\][/tex]
Thus, the values are:
[tex]\[
C_{12} = 9.0
\][/tex]
[tex]\[
C_{31} = 1.0
\][/tex]
[tex]\[
C_{22} = 8.0
\][/tex]
