Let's find the transpose of the given matrix step by step. The original matrix is:
[tex]\[
\begin{pmatrix}
3 & 10 & 7 \\
7 & 3 & 1 \\
2 & 7 & 10
\end{pmatrix}
\][/tex]
The transpose of a matrix is obtained by swapping its rows and columns. Here's how we can accomplish that:
1. Take the elements of the first column of the original matrix and place them as the first row of the transposed matrix.
2. Take the elements of the second column of the original matrix and place them as the second row of the transposed matrix.
3. Take the elements of the third column of the original matrix and place them as the third row of the transposed matrix.
Starting with the first column of the original matrix:
[tex]\[
\begin{pmatrix}
3 \\
7 \\
2
\end{pmatrix}
\][/tex]
This becomes the first row of the transposed matrix:
[tex]\[
\begin{pmatrix}
3 & 7 & 2
\end{pmatrix}
\][/tex]
Next, we take the second column of the original matrix:
[tex]\[
\begin{pmatrix}
10 \\
3 \\
7
\end{pmatrix}
\][/tex]
This becomes the second row of the transposed matrix:
[tex]\[
\begin{pmatrix}
3 & 7 & 2 \\
10 & 3 & 7
\end{pmatrix}
\][/tex]
Lastly, we take the third column of the original matrix:
[tex]\[
\begin{pmatrix}
7 \\
1 \\
10
\end{pmatrix}
\][/tex]
This becomes the third row of the transposed matrix:
[tex]\[
\begin{pmatrix}
3 & 7 & 2 \\
10 & 3 & 7 \\
7 & 1 & 10
\end{pmatrix}
\][/tex]
Therefore, the transpose of the matrix
[tex]\[
\begin{pmatrix}
3 & 10 & 7 \\
7 & 3 & 1 \\
2 & 7 & 10
\end{pmatrix}
\][/tex]
is
[tex]\[
\begin{pmatrix}
3 & 7 & 2 \\
10 & 3 & 7 \\
7 & 1 & 10
\end{pmatrix}
\][/tex]
So, the transpose of [tex]\(\begin{pmatrix}3 & 10 & 7 \\ 7 & 3 & 1 \\ 2 & 7 & 10\end{pmatrix}\)[/tex] is [tex]\(\begin{pmatrix}3 & 7 & 2 \\ 10 & 3 & 7 \\ 7 & 1 & 10\end{pmatrix}\)[/tex].
