To find the value of [tex]\(M + M^\top\)[/tex] where [tex]\(M\)[/tex] is given by
[tex]\[
M = \begin{pmatrix}
6 & 8 \\
5 & 4
\end{pmatrix},
\][/tex]
we need to follow these steps:
1. Determine the transpose of [tex]\(M\)[/tex]:
The transpose of a matrix, denoted as [tex]\(M^\top\)[/tex], is found by swapping the rows and columns. So, for the given matrix [tex]\(M\)[/tex],
[tex]\[
M^\top = \begin{pmatrix}
6 & 5 \\
8 & 4
\end{pmatrix}.
\][/tex]
2. Add [tex]\(M\)[/tex] and [tex]\(M^\top\)[/tex]:
To add two matrices, we simply add their corresponding elements. Therefore, we have:
[tex]\[
M + M^\top = \begin{pmatrix}
6 & 8 \\
5 & 4
\end{pmatrix}
+ \begin{pmatrix}
6 & 5 \\
8 & 4
\end{pmatrix}.
\][/tex]
Now, performing the element-wise addition:
[tex]\[
\begin{pmatrix}
6+6 & 8+5 \\
5+8 & 4+4
\end{pmatrix} = \begin{pmatrix}
12 & 13 \\
13 & 8
\end{pmatrix}.
\][/tex]
Thus, the resultant matrix [tex]\(M + M^\top\)[/tex] is:
[tex]\[
\begin{pmatrix}
12 & 13 \\
13 & 8
\end{pmatrix}.
\][/tex]
