Construct a `3xx3` matrix whose elements `a_(ij)` are given by
(i) `a_(ij)=i+j` (ii) `a_(ij)=ixxj` (iii) `a_(ij)=(i+j)^(2)`

Answer

To construct the `3x3` matrices based on the given formulas for the elements `a_(ij)`, we will follow these steps for each case:

### (i) For `a_(ij) = i + j`

1. **Identify the indices**: The indices `i` and `j` will take values from `1` to `3`.
2. **Calculate each element**:
- For `i = 1`:
- `a_(11) = 1 + 1 = 2`
- `a_(12) = 1 + 2 = 3`
- `a_(13) = 1 + 3 = 4`
- For `i = 2`:
- `a_(21) = 2 + 1 = 3`
- `a_(22) = 2 + 2 = 4`
- `a_(23) = 2 + 3 = 5`
- For `i = 3`:
- `a_(31) = 3 + 1 = 4`
- `a_(32) = 3 + 2 = 5`
- `a_(33) = 3 + 3 = 6`
3. **Construct the matrix**:
\[ A = \begin{bmatrix} 2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{bmatrix} \]

### (ii) For `a_(ij) = i * j`

1. **Identify the indices**: The indices `i` and `j` will again take values from `1` to `3`.
2. **Calculate each element**:
- For `i = 1`:
- `a_(11) = 1 * 1 = 1`
- `a_(12) = 1 * 2 = 2`
- `a_(13) = 1 * 3 = 3`
- For `i = 2`:
- `a_(21) = 2 * 1 = 2`
- `a_(22) = 2 * 2 = 4`
- `a_(23) = 2 * 3 = 6`
- For `i = 3`:
- `a_(31) = 3 * 1 = 3`
- `a_(32) = 3 * 2 = 6`
- `a_(33) = 3 * 3 = 9`
3. **Construct the matrix**:
\[ B = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix} \]

### (iii) For `a_(ij) = (i + j)^2`

1. **Identify the indices**: The indices `i` and `j` will take values from `1` to `3`.
2. **Calculate each element**:
- For `i = 1`:
- `a_(11) = (1 + 1)^2 = 2^2 = 4`
- `a_(12) = (1 + 2)^2 = 3^2 = 9`
- `a_(13) = (1 + 3)^2 = 4^2 = 16`
- For `i = 2`:
- `a_(21) = (2 + 1)^2 = 3^2 = 9`
- `a_(22) = (2 + 2)^2 = 4^2 = 16`
- `a_(23) = (2 + 3)^2 = 5^2 = 25`
- For `i = 3`:
- `a_(31) = (3 + 1)^2 = 4^2 = 16`
- `a_(32) = (3 + 2)^2 = 5^2 = 25`
- `a_(33) = (3 + 3)^2 = 6^2 = 36`
3. **Construct the matrix**:
\[ C = \begin{bmatrix} 4 & 9 & 16 \\ 9 & 16 & 25 \\ 16 & 25 & 36 \end{bmatrix} \]

### Summary of Matrices
- For `a_(ij) = i + j`:
\[ A = \begin{bmatrix} 2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{bmatrix} \]
- For `a_(ij) = i * j`:
\[ B = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix} \]
- For `a_(ij) = (i + j)^2`:
\[ C = \begin{bmatrix} 4 & 9 & 16 \\ 9 & 16 & 25 \\ 16 & 25 & 36 \end{bmatrix} \]