Consider a skew - symmetric matrix `A=[(a,b),(-b, c)]` such that a, b and c are selected from the set `S={0, 1, 2, 3,…………12}.` If `|A|` is divisible by 3, then the number of such possible matrices is

Options

A. 4
B. 5
C. 6
D. 12

Answer

To solve the problem, we will analyze the skew-symmetric matrix \( A \) and its determinant, and then determine how many values of \( b \) from the given set \( S \) make the determinant divisible by 3.

### Step-by-step Solution:

1. **Understanding the Skew-Symmetric Matrix**:
A skew-symmetric matrix has the property that its transpose is equal to its negative. For a 2x2 matrix of the form:
\[ A = \begin{pmatrix} a & b \\ -b & c \end{pmatrix} \]
the diagonal elements must be zero for it to be skew-symmetric. Therefore, we have:
\[ a = 0 \quad \text{and} \quad c = 0 \]
Thus, the matrix simplifies to:
\[ A = \begin{pmatrix} 0 & b \\ -b & 0 \end{pmatrix} \]

2. **Calculating the Determinant**:
The determinant of matrix \( A \) is calculated as follows:
\[ |A| = (0)(0) - (-b)(b) = 0 + b^2 = b^2 \]

3. **Condition for Divisibility by 3**:
We need to find when \( |A| = b^2 \) is divisible by 3. This means we need \( b^2 \equiv 0 \mod 3 \).

4. **Finding Values of \( b \)**:
We will check the values of \( b \) from the set \( S = \{0, 1, 2, 3, \ldots, 12\} \):
- \( b = 0 \): \( 0^2 = 0 \) (divisible by 3)
- \( b = 1 \): \( 1^2 = 1 \) (not divisible by 3)
- \( b = 2 \): \( 2^2 = 4 \) (not divisible by 3)
- \( b = 3 \): \( 3^2 = 9 \) (divisible by 3)
- \( b = 4 \): \( 4^2 = 16 \) (not divisible by 3)
- \( b = 5 \): \( 5^2 = 25 \) (not divisible by 3)
- \( b = 6 \): \( 6^2 = 36 \) (divisible by 3)
- \( b = 7 \): \( 7^2 = 49 \) (not divisible by 3)
- \( b = 8 \): \( 8^2 = 64 \) (not divisible by 3)
- \( b = 9 \): \( 9^2 = 81 \) (divisible by 3)
- \( b = 10 \): \( 10^2 = 100 \) (not divisible by 3)
- \( b = 11 \): \( 11^2 = 121 \) (not divisible by 3)
- \( b = 12 \): \( 12^2 = 144 \) (divisible by 3)

5. **Counting the Valid Values**:
The values of \( b \) that make \( b^2 \) divisible by 3 are:
- \( b = 0 \)
- \( b = 3 \)
- \( b = 6 \)
- \( b = 9 \)
- \( b = 12 \)

Thus, there are a total of 5 valid values of \( b \).

### Final Answer:
The number of such possible matrices is **5**.