let A `={{:({:(a_(11),a_(12)),(a_(21),a_(22)):}):a_(ij){0,1,2} and a_(11)=a_(22)}`
then the number of singular matrices in set A is

Options

A. 7
B. 10
C. 5
D. 3

Explanation

To solve the problem, we need to determine the number of singular matrices in the set \( A \) defined by the conditions given.

### Step-by-step Solution:

1. **Understanding the Matrix Structure**:
The matrix \( A \) is defined as:
\[ A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \]
where \( a_{ij} \in \{0, 1, 2\} \) and \( a_{11} = a_{22} \).

2. **Finding the Determinant**:
The determinant of matrix \( A \) is given by:
\[ \text{det}(A) = a_{11} \cdot a_{22} - a_{12} \cdot a_{21} \]
Since \( a_{11} = a_{22} \), we can rewrite the determinant as:
\[ \text{det}(A) = a_{11}^2 - a_{12} \cdot a_{21} \]

3. **Condition for Singular Matrix**:
A matrix is singular if its determinant is zero:
\[ a_{11}^2 - a_{12} \cdot a_{21} = 0 \]
This implies:
\[ a_{11}^2 = a_{12} \cdot a_{21} \]

4. **Case Analysis**:
We will analyze different cases based on the possible values of \( a_{11} \).

- **Case 1**: \( a_{11} = 0 \)
\[ 0^2 = a_{12} \cdot a_{21} \implies a_{12} \cdot a_{21} = 0 \]
This means either \( a_{12} = 0 \) or \( a_{21} = 0 \). The combinations are:
- \( (a_{12}, a_{21}) = (0, 0) \) → 1 way
- \( (a_{12}, a_{21}) = (1, 0) \) → 1 way
- \( (a_{12}, a_{21}) = (2, 0) \) → 1 way
- \( (a_{12}, a_{21}) = (0, 1) \) → 1 way
- \( (a_{12}, a_{21}) = (0, 2) \) → 1 way
This gives us a total of 5 ways.

- **Case 2**: \( a_{11} = 1 \)
\[ 1^2 = a_{12} \cdot a_{21} \implies a_{12} \cdot a_{21} = 1 \]
The valid pairs are:
- \( (a_{12}, a_{21}) = (1, 1) \) → 1 way
This gives us a total of 1 way.

- **Case 3**: \( a_{11} = 2 \)
\[ 2^2 = a_{12} \cdot a_{21} \implies 4 = a_{12} \cdot a_{21} \]
The valid pairs are:
- \( (a_{12}, a_{21}) = (2, 2) \) → 1 way
This gives us a total of 1 way.

5. **Total Count of Singular Matrices**:
Adding all the cases together:
\[ \text{Total} = 5 \text{ (from Case 1)} + 1 \text{ (from Case 2)} + 1 \text{ (from Case 3)} = 7 \]

### Final Answer:
The total number of singular matrices in set \( A \) is **7**.