To find the elements of the matrices [tex]\(A + B\)[/tex] and [tex]\(A - B\)[/tex], we need to perform matrix addition and subtraction.
Matrix [tex]\(A\)[/tex] and matrix [tex]\(B\)[/tex] are given as:
[tex]\[
A = \begin{pmatrix}
-3 & 1 & 0 & 7 \\
5 & -6 & 1 & 3 \\
8 & 2 & 6 & -5
\end{pmatrix}
\][/tex]
[tex]\[
B = \begin{pmatrix}
5 & 3 & -7 & -5 \\
0 & 1 & 6 & -6 \\
5 & -4 & 9 & 3
\end{pmatrix}
\][/tex]
### Matrix Addition [tex]\(A + B\)[/tex]:
To add two matrices, we add their corresponding elements:
[tex]\[
A + B = \begin{pmatrix}
-3 + 5 & 1 + 3 & 0 + (-7) & 7 + (-5) \\
5 + 0 & -6 + 1 & 1 + 6 & 3 + (-6) \\
8 + 5 & 2 + (-4) & 6 + 9 & -5 + 3
\end{pmatrix}
\][/tex]
[tex]\[
A + B = \begin{pmatrix}
2 & 4 & -7 & 2 \\
5 & -5 & 7 & -3 \\
13 & -2 & 15 & -2
\end{pmatrix}
\][/tex]
Here, [tex]\(x_{24}\)[/tex] represents the element in the 2nd row and 4th column of [tex]\(A + B\)[/tex]. So,
[tex]\[
x_{24} = -3
\][/tex]
### Matrix Subtraction [tex]\(A - B\)[/tex]:
To subtract two matrices, we subtract their corresponding elements:
[tex]\[
A - B = \begin{pmatrix}
-3 - 5 & 1 - 3 & 0 - (-7) & 7 - (-5) \\
5 - 0 & -6 - 1 & 1 - 6 & 3 - (-6) \\
8 - 5 & 2 - (-4) & 6 - 9 & -5 - 3
\end{pmatrix}
\][/tex]
[tex]\[
A - B = \begin{pmatrix}
-8 & -2 & 7 & 12 \\
5 & -7 & -5 & 9 \\
3 & 6 & -3 & -8
\end{pmatrix}
\][/tex]
Here, [tex]\(y_{13}\)[/tex] represents the element in the 1st row and 3rd column of [tex]\(A - B\)[/tex]. So,
[tex]\[
y_{13} = 7
\][/tex]
Therefore, [tex]\(x_{24}\)[/tex] represents the element [tex]\(-3\)[/tex] and [tex]\(y_{13}\)[/tex] represents the element [tex]\(7\)[/tex].
