Find the number of `2xx2` matrix satisfying
(i) aij is 1 or -1
(ii) `a_(11)^(2)+a_(12)^(2)=a_(21)^(2)+a_(22)^(2)=2`
(iii) `a_(11)a_(21)+a_(12)a_(22)=0`

Answer

To solve the problem, we need to find the number of \(2 \times 2\) matrices \(A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}\) that satisfy the following conditions:

1. Each element \(a_{ij}\) is either \(1\) or \(-1\).
2. \(a_{11}^2 + a_{12}^2 = 2\) and \(a_{21}^2 + a_{22}^2 = 2\).
3. \(a_{11}a_{21} + a_{12}a_{22} = 0\).

### Step 1: Analyzing the first condition
Since each element \(a_{ij}\) can be either \(1\) or \(-1\), we have:
- \(a_{11}^2 = 1\) if \(a_{11} = 1\) or \(-1\).
- \(a_{12}^2 = 1\) if \(a_{12} = 1\) or \(-1\).
- Similarly for \(a_{21}\) and \(a_{22}\).

Thus, the condition \(a_{11}^2 + a_{12}^2 = 2\) implies that both \(a_{11}\) and \(a_{12}\) must be non-zero, meaning they can only be \(1\) or \(-1\). The same applies to \(a_{21}\) and \(a_{22}\).

### Step 2: Finding combinations for \(a_{11}\) and \(a_{12}\)
The only combinations of \(a_{11}\) and \(a_{12}\) that satisfy \(a_{11}^2 + a_{12}^2 = 2\) are:
- \(a_{11} = 1, a_{12} = 1\)
- \(a_{11} = -1, a_{12} = -1\)
- \(a_{11} = 1, a_{12} = -1\)
- \(a_{11} = -1, a_{12} = 1\)

Thus, there are \(4\) valid combinations for the first row.

### Step 3: Finding combinations for \(a_{21}\) and \(a_{22}\)
Similarly, for \(a_{21}\) and \(a_{22}\), the valid combinations are also \(4\):
- \(a_{21} = 1, a_{22} = 1\)
- \(a_{21} = -1, a_{22} = -1\)
- \(a_{21} = 1, a_{22} = -1\)
- \(a_{21} = -1, a_{22} = 1\)

### Step 4: Applying the third condition
Now we need to satisfy the condition \(a_{11}a_{21} + a_{12}a_{22} = 0\). This means that the product of the elements in the first column and the second column must cancel each other out.

Let's analyze the combinations:
1. If \(a_{11} = 1\) and \(a_{12} = 1\):
- \(a_{21} = 1\) and \(a_{22} = -1\) gives \(1 \cdot 1 + 1 \cdot (-1) = 0\).
- \(a_{21} = -1\) and \(a_{22} = 1\) gives \(1 \cdot (-1) + 1 \cdot 1 = 0\).
- Total valid combinations: \(2\).

2. If \(a_{11} = -1\) and \(a_{12} = -1\):
- \(a_{21} = 1\) and \(a_{22} = -1\) gives \(-1 \cdot 1 + (-1) \cdot (-1) = 0\).
- \(a_{21} = -1\) and \(a_{22} = 1\) gives \(-1 \cdot (-1) + (-1) \cdot 1 = 0\).
- Total valid combinations: \(2\).

3. If \(a_{11} = 1\) and \(a_{12} = -1\):
- \(a_{21} = 1\) and \(a_{22} = 1\) gives \(1 \cdot 1 + (-1) \cdot 1 \neq 0\) (not valid).
- \(a_{21} = -1\) and \(a_{22} = -1\) gives \(1 \cdot (-1) + (-1) \cdot (-1) \neq 0\) (not valid).
- Total valid combinations: \(0\).

4. If \(a_{11} = -1\) and \(a_{12} = 1\):
- \(a_{21} = 1\) and \(a_{22} = -1\) gives \(-1 \cdot 1 + 1 \cdot (-1) \neq 0\) (not valid).
- \(a_{21} = -1\) and \(a_{22} = 1\) gives \(-1 \cdot (-1) + 1 \cdot 1 \neq 0\) (not valid).
- Total valid combinations: \(0\).

### Step 5: Total valid matrices
Adding the valid combinations from all cases:
- From the first case: \(2\)
- From the second case: \(2\)
- From the third case: \(0\)
- From the fourth case: \(0\)

Thus, the total number of valid \(2 \times 2\) matrices is \(2 + 2 + 0 + 0 = 4\).

### Final Answer
The total number of \(2 \times 2\) matrices satisfying the given conditions is **4**.