If matrix `A=[a_(ij)]_(3xx2)` and `a_(ij)=(3i-2j)^(2)`, then find matrix A.

Answer

To find the matrix \( A = [a_{ij}]_{3 \times 2} \) where \( a_{ij} = (3i - 2j)^2 \), we will calculate each element of the matrix based on the given formula.

### Step-by-Step Solution:

1. **Identify the dimensions of the matrix**:
- The matrix \( A \) is of order \( 3 \times 2 \), which means it has 3 rows and 2 columns.

2. **Determine the indices**:
- The row indices \( i \) can take values 1, 2, and 3.
- The column indices \( j \) can take values 1 and 2.

3. **Calculate each element of the matrix**:
- **Row 1**:
- \( a_{11} = (3 \cdot 1 - 2 \cdot 1)^2 = (3 - 2)^2 = 1^2 = 1 \)
- \( a_{12} = (3 \cdot 1 - 2 \cdot 2)^2 = (3 - 4)^2 = (-1)^2 = 1 \)

- **Row 2**:
- \( a_{21} = (3 \cdot 2 - 2 \cdot 1)^2 = (6 - 2)^2 = 4^2 = 16 \)
- \( a_{22} = (3 \cdot 2 - 2 \cdot 2)^2 = (6 - 4)^2 = 2^2 = 4 \)

- **Row 3**:
- \( a_{31} = (3 \cdot 3 - 2 \cdot 1)^2 = (9 - 2)^2 = 7^2 = 49 \)
- \( a_{32} = (3 \cdot 3 - 2 \cdot 2)^2 = (9 - 4)^2 = 5^2 = 25 \)

4. **Construct the matrix**:
- Now that we have all the elements, we can write the matrix \( A \):
\[ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 16 & 4 \\ 49 & 25 \end{bmatrix} \]

### Final Answer:
The matrix \( A \) is:
\[ A = \begin{bmatrix} 1 & 1 \\ 16 & 4 \\ 49 & 25 \end{bmatrix} \]