If the elements of a matrix A of order `2 xx 3` are defined as `a_(ij) = {{:(i + j , i =j),(i-j,i nej):}` then the matrix `A^(T)` is :

Options

A. `[ (-2 ,1 ,2),(-1,-4,1)]`
B. `[(2,1),(-1,4),(-2,-1)]`
C. `[(2,-1,-2),(1,4,-1)]`
D. `[(0,3),(3,0),(4,5)]`

Answer

To solve the problem, we need to find the transpose of the matrix \( A \) defined by the given conditions. Let's break down the steps to find the matrix \( A \) and then its transpose \( A^T \).

### Step 1: Define the matrix \( A \)

The matrix \( A \) is of order \( 2 \times 3 \), which means it has 2 rows and 3 columns. The elements of the matrix are defined as follows:

- If \( i = j \), then \( a_{ij} = i + j \)
- If \( i \neq j \), then \( a_{ij} = i - j \)

### Step 2: Calculate the elements of \( A \)

We will calculate each element of the matrix \( A \):

1. **For \( i = 1 \)**:
- \( j = 1 \): \( a_{11} = 1 + 1 = 2 \)
- \( j = 2 \): \( a_{12} = 1 - 2 = -1 \)
- \( j = 3 \): \( a_{13} = 1 - 3 = -2 \)

2. **For \( i = 2 \)**:
- \( j = 1 \): \( a_{21} = 2 - 1 = 1 \)
- \( j = 2 \): \( a_{22} = 2 + 2 = 4 \)
- \( j = 3 \): \( a_{23} = 2 - 3 = -1 \)

Now we can write the matrix \( A \):

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

### Step 3: Find the transpose of \( A \)

The transpose of a matrix \( A \), denoted as \( A^T \), is obtained by swapping its rows and columns. Therefore, the first row of \( A \) becomes the first column of \( A^T \), and the second row of \( A \) becomes the second column of \( A^T \).

So, we can write:

\[ A^T = \begin{pmatrix} 2 & 1 \\ -1 & 4 \\ -2 & -1 \end{pmatrix} \]

### Final Result

Thus, the transpose of the matrix \( A \) is:

\[ A^T = \begin{pmatrix} 2 & 1 \\ -1 & 4 \\ -2 & -1 \end{pmatrix} \]