To find the elements [tex]\(C_{12}\)[/tex], [tex]\(C_{31}\)[/tex], and [tex]\(C_{22}\)[/tex] of the matrix [tex]\(C\)[/tex] that is obtained by summing two matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex], let's go through the steps in detail:
1. Define the matrices:
[tex]\[
A = \begin{bmatrix}
16 & 9 \\
-3 & 0 \\
4 & -10
\end{bmatrix}
\][/tex]
[tex]\[
B = \begin{bmatrix}
-0.5 & 0 \\
5 & 8 \\
-3 & 14
\end{bmatrix}
\][/tex]
2. Sum the matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex] to get [tex]\(C\)[/tex]:
[tex]\[
C = A + B = \begin{bmatrix}
16 & 9 \\
-3 & 0 \\
4 & -10
\end{bmatrix} + \begin{bmatrix}
-0.5 & 0 \\
5 & 8 \\
-3 & 14
\end{bmatrix}
\][/tex]
This results in:
[tex]\[
C = \begin{bmatrix}
16 + (-0.5) & 9 + 0 \\
-3 + 5 & 0 + 8 \\
4 + (-3) & -10 + 14
\end{bmatrix}
\][/tex]
Simplifying the elements inside matrix [tex]\(C\)[/tex]:
[tex]\[
C = \begin{bmatrix}
15.5 & 9 \\
2 & 8 \\
1 & 4
\end{bmatrix}
\][/tex]
3. Extract the specific elements [tex]\(C_{12}\)[/tex], [tex]\(C_{31}\)[/tex], and [tex]\(C_{22}\)[/tex] from matrix [tex]\(C\)[/tex]:
- [tex]\(C_{12}\)[/tex]: This element is located at the first row and second column of [tex]\(C\)[/tex].
[tex]\[
C_{12} = 9
\][/tex]
- [tex]\(C_{31}\)[/tex]: This element is located at the third row and first column of [tex]\(C\)[/tex].
[tex]\[
C_{31} = 1
\][/tex]
- [tex]\(C_{22}\)[/tex]: This element is located at the second row and second column of [tex]\(C\)[/tex].
[tex]\[
C_{22} = 8
\][/tex]
Therefore, the elements [tex]\(C_{12}\)[/tex], [tex]\(C_{31}\)[/tex], and [tex]\(C_{22}\)[/tex] of matrix [tex]\(C\)[/tex] are:
[tex]\[
C_{12} = 9, \quad C_{31} = 1, \quad C_{22} = 8
\][/tex]
