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 solve the problem of finding the number of singular matrices \( A = \begin{pmatrix} a & b \\ c & a \end{pmatrix} \) where \( a, b, c \in \{0, 1, 2\} \), we need to follow these steps:

### Step 1: Understand the condition for singularity
A matrix is singular if its determinant is zero. The determinant of matrix \( A \) is given by:
\[ \text{det}(A) = a \cdot a - b \cdot c = a^2 - bc \]
For \( A \) to be singular, we need:
\[ a^2 - bc = 0 \implies a^2 = bc \]

### Step 2: Analyze the possible values of \( a, b, c \)
Since \( a, b, c \) can take values from the set \( \{0, 1, 2\} \), we will analyze each possible value of \( a \) and determine the corresponding values of \( b \) and \( c \) that satisfy the equation \( bc = a^2 \).

### Step 3: Case Analysis for \( a \)

#### Case 1: \( a = 0 \)
\[ a^2 = 0^2 = 0 \implies bc = 0 \]
To satisfy \( bc = 0 \), either \( b = 0 \) or \( c = 0 \):
- If \( b = 0 \), then \( c \) can be \( 0, 1, 2 \) (3 options).
- If \( c = 0 \), then \( b \) can be \( 0, 1, 2 \) (3 options).
However, the case \( b = 0, c = 0 \) is counted twice, so we have:
\[ 3 + 3 - 1 = 5 \text{ options} \]

#### Case 2: \( a = 1 \)
\[ a^2 = 1^2 = 1 \implies bc = 1 \]
To satisfy \( bc = 1 \), the pairs \( (b, c) \) can be:
- \( (1, 1) \) (1 option)

#### Case 3: \( a = 2 \)
\[ a^2 = 2^2 = 4 \implies bc = 4 \]
To satisfy \( bc = 4 \), the pairs \( (b, c) \) can be:
- \( (2, 2) \) (1 option)

### Step 4: Summing the possibilities
Now we sum the number of valid pairs for each case:
- From \( a = 0 \): 5 options
- From \( a = 1 \): 1 option
- From \( a = 2 \): 1 option

Total number of singular matrices \( A \):
\[ 5 + 1 + 1 = 7 \]

### Final Answer
The number of possible singular matrices \( A \) is \( \boxed{7} \).