To solve the problem, let's follow these steps in detail:
1. Identify the given matrices: We start with the matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Matrix [tex]\( A \)[/tex] is:
[tex]\[
A = \begin{pmatrix}
1 & 2 \\
3 & 4
\end{pmatrix}
\][/tex]
Matrix [tex]\( B \)[/tex] is:
[tex]\[
B = \begin{pmatrix}
1 & 2 \\
0 & 3
\end{pmatrix}
\][/tex]
2. Find the transpose of matrix [tex]\( A \)[/tex]: The transpose of a matrix is obtained by flipping it over its diagonal, which means switching the row and column indices of the matrix.
So, the transpose of [tex]\( A \)[/tex] (denoted as [tex]\( A^\top \)[/tex]) is:
[tex]\[
A^\top = \begin{pmatrix}
1 & 3 \\
2 & 4
\end{pmatrix}
\][/tex]
3. Add the transpose of [tex]\( A \)[/tex] to matrix [tex]\( B \)[/tex]: Matrix addition involves adding corresponding elements from each matrix.
Adding [tex]\( A^\top \)[/tex] and [tex]\( B \)[/tex] element-wise:
[tex]\[
A^\top + B = \begin{pmatrix}
1 & 3 \\
2 & 4
\end{pmatrix} + \begin{pmatrix}
1 & 2 \\
0 & 3
\end{pmatrix}
\][/tex]
To perform this addition:
- Add the elements in the first row, first column: [tex]\( 1 + 1 = 2 \)[/tex]
- Add the elements in the first row, second column: [tex]\( 3 + 2 = 5 \)[/tex]
- Add the elements in the second row, first column: [tex]\( 2 + 0 = 2 \)[/tex]
- Add the elements in the second row, second column: [tex]\( 4 + 3 = 7 \)[/tex]
Thus, the result is:
[tex]\[
A^\top + B = \begin{pmatrix}
2 & 5 \\
2 & 7
\end{pmatrix}
\][/tex]
Therefore, the final result of adding the transpose of [tex]\( A \)[/tex] to [tex]\( B \)[/tex] is:
[tex]\[
\begin{pmatrix}
2 & 5 \\
2 & 7
\end{pmatrix}
\][/tex]
