If A = (34221−1) ( 3 2 1 4 2 − 1 ) and B = ⎛⎝⎜103412⎞⎠⎟ ( 1 4 0 1 3 2 ) . Find AT T + B, ( where T means transpose)

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)