Construct a matrix `[a_(ij)]_(3xx3)` , where `a_(ij)=2i-3j`.

Explanation

To construct the matrix \( A = [a_{ij}]_{3 \times 3} \) where \( a_{ij} = 2i - 3j \), we will follow these steps:

### Step 1: Define the matrix structure
Since we need a \( 3 \times 3 \) matrix, we will have rows and columns indexed by \( i \) and \( j \) respectively, where \( i, j = 1, 2, 3 \).

### Step 2: Calculate the elements of the matrix
We will compute each element \( a_{ij} \) using the formula \( a_{ij} = 2i - 3j \).

#### Row 1:
- For \( a_{11} \):
\[ a_{11} = 2(1) - 3(1) = 2 - 3 = -1 \]
- For \( a_{12} \):
\[ a_{12} = 2(1) - 3(2) = 2 - 6 = -4 \]
- For \( a_{13} \):
\[ a_{13} = 2(1) - 3(3) = 2 - 9 = -7 \]

#### Row 2:
- For \( a_{21} \):
\[ a_{21} = 2(2) - 3(1) = 4 - 3 = 1 \]
- For \( a_{22} \):
\[ a_{22} = 2(2) - 3(2) = 4 - 6 = -2 \]
- For \( a_{23} \):
\[ a_{23} = 2(2) - 3(3) = 4 - 9 = -5 \]

#### Row 3:
- For \( a_{31} \):
\[ a_{31} = 2(3) - 3(1) = 6 - 3 = 3 \]
- For \( a_{32} \):
\[ a_{32} = 2(3) - 3(2) = 6 - 6 = 0 \]
- For \( a_{33} \):
\[ a_{33} = 2(3) - 3(3) = 6 - 9 = -3 \]

### Step 3: Assemble the matrix
Now that we have all the elements, we can write the matrix \( A \):
\[ A = \begin{bmatrix} -1 & -4 & -7 \\ 1 & -2 & -5 \\ 3 & 0 & -3 \end{bmatrix} \]

### Final Matrix
Thus, the constructed matrix is:
\[ A = \begin{bmatrix} -1 & -4 & -7 \\ 1 & -2 & -5 \\ 3 & 0 & -3 \end{bmatrix} \]