Let `A=[(a, b),(c, a)]AA a, b, c, in {0, 1, 2}`. If A is a singular matrix, then the number of possible matrices A are

Options

A. 18
B. 27
C. 7
D. 3

Answer

To determine the number of possible matrices \( A = \begin{pmatrix} a & b \\ c & a \end{pmatrix} \) that are singular, we need to analyze the condition for a matrix to be singular. A matrix is singular if its determinant is zero.

### Step 1: Calculate the determinant of matrix \( A \)

The determinant of matrix \( A \) is given by:

\[ \text{det}(A) = a \cdot a - b \cdot c = a^2 - bc \]

### Step 2: Set the determinant to zero

For the matrix \( A \) to be singular, we set the determinant equal to zero:

\[ a^2 - bc = 0 \]

This implies:

\[ a^2 = bc \]

### Step 3: Consider the values of \( a, b, c \)

Given that \( a, b, c \) can take values from the set \( \{0, 1, 2\} \), we will analyze the possible combinations of \( a, b, c \) that satisfy the equation \( a^2 = bc \).

### Step 4: Analyze cases based on the value of \( a \)

1. **Case 1: \( a = 0 \)**

\[ 0^2 = bc \implies 0 = bc \]

Here, either \( b = 0 \) or \( c = 0 \) (or both). The pairs \( (b, c) \) can be:
- \( (0, 0) \)
- \( (0, 1) \)
- \( (0, 2) \)
- \( (1, 0) \)
- \( (2, 0) \)

This gives us **5 pairs**.

2. **Case 2: \( a = 1 \)**

\[ 1^2 = bc \implies 1 = bc \]

The pairs \( (b, c) \) that satisfy \( bc = 1 \) are:
- \( (1, 1) \)

This gives us **1 pair**.

3. **Case 3: \( a = 2 \)**

\[ 2^2 = bc \implies 4 = bc \]

The pairs \( (b, c) \) that satisfy \( bc = 4 \) are:
- \( (2, 2) \)

This gives us **1 pair**.

### Step 5: Sum the total pairs

Now, we sum the number of pairs from all cases:

- From Case 1: 5 pairs
- From Case 2: 1 pair
- From Case 3: 1 pair

Total pairs = \( 5 + 1 + 1 = 7 \)

### Conclusion

The total number of possible matrices \( A \) that are singular is **7**.