The given expression \( a_{ij} = 2i + j \) defines the elements of a matrix \( A \) where \( i \) and \( j \) are the row and column indices, respectively. Each element of the matrix is determined by plugging the values of \( i \) (row index) and \( j \) (column index) into the formula.

Let's consider an example to illustrate how to compute the elements of the matrix:

### Example: 3x3 Matrix
Suppose we want to create a \( 3 \times 3 \) matrix using the given formula.

\[
A = \begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}
\]

Now, calculate each element using the formula \( a_{ij} = 2i + j \):

- \( a_{11} = 2(1) + 1 = 2 + 1 = 3 \)
- \( a_{12} = 2(1) + 2 = 2 + 2 = 4 \)
- \( a_{13} = 2(1) + 3 = 2 + 3 = 5 \)
- \( a_{21} = 2(2) + 1 = 4 + 1 = 5 \)
- \( a_{22} = 2(2) + 2 = 4 + 2 = 6 \)
- \( a_{23} = 2(2) + 3 = 4 + 3 = 7 \)
- \( a_{31} = 2(3) + 1 = 6 + 1 = 7 \)
- \( a_{32} = 2(3) + 2 = 6 + 2 = 8 \)
- \( a_{33} = 2(3) + 3 = 6 + 3 = 9 \)

So, the resulting matrix \( A \) is:

\[
A = \begin{pmatrix}
3 & 4 & 5 \\
5 & 6 & 7 \\
7 & 8 & 9
\end{pmatrix}
\]

### Would you like more details or have any questions?

Here are some related questions you might consider:
1. How would the matrix look for a \( 4 \times 4 \) matrix using the same formula?
2. What would happen if the formula was \( a_{ij} = 2i - j \)?
3. How does the pattern change if the formula is \( a_{ij} = i^2 + j \)?
4. Can you generalize the pattern observed in the matrix elements for any \( m \times n \) matrix?
5. How would the matrix change if both \( i \) and \( j \) started from 0 instead of 1?

**Tip:** When dealing with matrix problems, carefully consider how the indices \( i \) and \( j \) influence the pattern of elements, as this can help you quickly identify or verify matrix properties.

Explore how to calculate matrix elements using the formula a_{ij} = 2i + j. This example illustrates the application of matrix operations in creating a 3x3 matrix, covering row and column indices. Suitable for high school students and those interested in matrix algebra.

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