If matrix `A=[{:(,2,1,3),(,4,-3,2):}] and B=[{:(,3,-2),(,7,4):}]`, find transpose matrices `A^t and B^t`. If possible, find (i) `A+A^t` (ii) `B+B^t`

Answer

To solve the problem, we will follow these steps:

### Step 1: Find the transpose of matrix A

Given:
\[ A = \begin{pmatrix} 2 & 1 & 3 \\ 4 & -3 & 2 \end{pmatrix} \]

To find the transpose \( A^t \), we convert the rows of \( A \) into columns:
- The first row \( (2, 1, 3) \) becomes the first column of \( A^t \).
- The second row \( (4, -3, 2) \) becomes the second column of \( A^t \).

Thus, the transpose of matrix A is:
\[ A^t = \begin{pmatrix} 2 & 4 \\ 1 & -3 \\ 3 & 2 \end{pmatrix} \]

### Step 2: Find the transpose of matrix B

Given:
\[ B = \begin{pmatrix} 3 & -2 \\ 7 & 4 \end{pmatrix} \]

To find the transpose \( B^t \), we convert the rows of \( B \) into columns:
- The first row \( (3, -2) \) becomes the first column of \( B^t \).
- The second row \( (7, 4) \) becomes the second column of \( B^t \).

Thus, the transpose of matrix B is:
\[ B^t = \begin{pmatrix} 3 & 7 \\ -2 & 4 \end{pmatrix} \]

### Step 3: Find \( A + A^t \)

To find \( A + A^t \), we need to check the dimensions of both matrices:
- \( A \) is a \( 2 \times 3 \) matrix.
- \( A^t \) is a \( 3 \times 2 \) matrix.

Since the dimensions do not match (they cannot be added), we conclude:
\[ A + A^t \text{ is not possible.} \]

### Step 4: Find \( B + B^t \)

To find \( B + B^t \), we check the dimensions:
- \( B \) is a \( 2 \times 2 \) matrix.
- \( B^t \) is also a \( 2 \times 2 \) matrix.

Since the dimensions match, we can add them:
\[ B + B^t = \begin{pmatrix} 3 & -2 \\ 7 & 4 \end{pmatrix} + \begin{pmatrix} 3 & 7 \\ -2 & 4 \end{pmatrix} \]

Now, we add the corresponding elements:
- First element: \( 3 + 3 = 6 \)
- Second element: \( -2 + 7 = 5 \)
- Third element: \( 7 - 2 = 5 \)
- Fourth element: \( 4 + 4 = 8 \)

Thus, we get:
\[ B + B^t = \begin{pmatrix} 6 & 5 \\ 5 & 8 \end{pmatrix} \]

### Final Results
- \( A^t = \begin{pmatrix} 2 & 4 \\ 1 & -3 \\ 3 & 2 \end{pmatrix} \)
- \( B^t = \begin{pmatrix} 3 & 7 \\ -2 & 4 \end{pmatrix} \)
- \( A + A^t \) is not possible.
- \( B + B^t = \begin{pmatrix} 6 & 5 \\ 5 & 8 \end{pmatrix} \)