To find the value of [tex]\(A + B + C\)[/tex], where:
[tex]\[ A = \begin{bmatrix} 4 & 5 & 7 \\ 3 & 8 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 6 & 4 & -2 \\ 5 & 3 & -6 \end{bmatrix}, \quad C = \begin{bmatrix} 3 & -2 & 1 \\ 4 & 3 & -2 \end{bmatrix} \][/tex]
we need to follow these steps:
1. Add the corresponding elements of matrix [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex].
Let's add the elements in the first row:
[tex]\[
\begin{array}{ccc}
A_{11} + B_{11} + C_{11} & A_{12} + B_{12} + C_{12} & A_{13} + B_{13} + C_{13} \\
4 + 6 + 3 & 5 + 4 - 2 & 7 - 2 + 1
\end{array}
\][/tex]
Which results in:
[tex]\[
\begin{array}{ccc}
13 & 7 & 6
\end{array}
\][/tex]
Now let's add the elements in the second row:
[tex]\[
\begin{array}{ccc}
A_{21} + B_{21} + C_{21} & A_{22} + B_{22} + C_{22} & A_{23} + B_{23} + C_{23} \\
3 + 5 + 4 & 8 + 3 + 3 & 2 - 6 - 2
\end{array}
\][/tex]
Which results in:
[tex]\[
\begin{array}{ccc}
12 & 14 & -6
\end{array}
\][/tex]
2. Form the resultant matrix.
Combining these results, the sum of [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] is:
[tex]\[
A + B + C = \begin{bmatrix} 13 & 7 & 6 \\ 12 & 14 & -6 \end{bmatrix}
\][/tex]
3. Find the value—sum of all elements in the resultant matrix.
To find the value of [tex]\(A + B + C\)[/tex], sum all the elements of the resultant matrix:
[tex]\[
13 + 7 + 6 + 12 + 14 - 6
\][/tex]
Combine these values:
[tex]\[
13 + 7 = 20
\][/tex]
[tex]\[
20 + 6 = 26
\][/tex]
[tex]\[
26 + 12 = 38
\][/tex]
[tex]\[
38 + 14 = 52
\][/tex]
[tex]\[
52 - 6 = 46
\][/tex]
Thus, the value of [tex]\(A + B + C\)[/tex] is 46.
