To find the expression \(A^T + B\), where \(A\) and \(B\) are given matrices, we must first transpose matrix \(A\) (denoted \(A^T\)) and then add it to matrix \(B\).
Step 1: Transpose Matrix A
Matrix \(A\) is given as:
\[A = \begin{bmatrix} 3 & 4 \\ 2 & 2 \\ 1 & -1 \end{bmatrix}\]
To transpose \(A\), we swap its rows with columns:
\[A^T = \begin{bmatrix} 3 & 2 & 1 \\ 4 & 2 & -1 \end{bmatrix}\]
Step 2: Add the Transposed Matrix \(A^T\) with Matrix B
Matrix \(B\) is given as:
\[B = \begin{bmatrix} 1 & 4 \\ 0 & 1 \\ 3 & 2 \end{bmatrix}\]
Now, add \(A^T\) to \(B\):
\[ A^T + B = \begin{bmatrix} 3 & 2 & 1 \\ 4 & 2 & -1 \end{bmatrix} + \begin{bmatrix} 1 & 4 \\ 0 & 1 \\ 3 & 2 \end{bmatrix} \]
Adding corresponding elements, we have:
\[ = \begin{bmatrix} (3+1) & (2+4) \\ (4+0) & (2+1) \\ (1+3) & (-1+2) \end{bmatrix} \]
Simplifying the additions, we get:
\[A^T + B = \begin{bmatrix} 4 & 6 \\ 4 & 3 \\ 4 & 1 \end{bmatrix}\]
The correct option for \(A^T + B\) is: (424831)