The given expression \( a_{ij} = (-1)^{i+j} \) defines the elements of a matrix \( A \), where each element depends on the sum of the indices \( i \) and \( j \).

### Key Points:
1. **Element Definition**: For any element \( a_{ij} \) in the matrix, its value is determined by the formula \( a_{ij} = (-1)^{i+j} \).
   - If \( i + j \) is **even**, then \( (-1)^{i+j} = 1 \).
   - If \( i + j \) is **odd**, then \( (-1)^{i+j} = -1 \).

2. **Pattern of the Matrix**: The matrix will have a checkerboard pattern of \( 1 \)'s and \( -1 \)'s. Specifically:
   - For example, in a 3x3 matrix:
   \[
   A = \begin{pmatrix}
   1 & -1 & 1 \\
   -1 & 1 & -1 \\
   1 & -1 & 1 \\
   \end{pmatrix}
   \]

### Example Calculation:
- \( a_{11} = (-1)^{1+1} = (-1)^2 = 1 \)
- \( a_{12} = (-1)^{1+2} = (-1)^3 = -1 \)
- \( a_{21} = (-1)^{2+1} = (-1)^3 = -1 \)
- \( a_{22} = (-1)^{2+2} = (-1)^4 = 1 \)

### Summary:
The expression \( a_{ij} = (-1)^{i+j} \) results in a matrix with alternating signs depending on the sum of the indices. This kind of matrix is useful in various mathematical contexts, including determinants and linear algebra problems.

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Do you want more details or have any questions?

Here are 5 related questions you might explore:
1. How would you construct a 4x4 matrix using this formula?
2. What is the determinant of a 2x2 matrix generated using this formula?
3. How does this pattern affect the eigenvalues of the matrix?
4. Can you generalize this pattern for non-square matrices?
5. How does this matrix behave when multiplied by itself?

**Tip:** Understanding patterns in matrix elements can simplify complex matrix operations, especially in linear algebra.

Explore the checkerboard pattern matrix generated by the formula a_{ij} = (-1)^{i+j}. Understand how the sum of indices affects the matrix elements, leading to a pattern of 1's and -1's. This concept is key in matrix algebra and suitable for students in Grades 9-12.

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