Let p be an odd prime number and `T_p`, be the following set of `2 xx 2` matrices `T_p={A=[(a,b),(c,a)]:a,b,c in {0,1,2,.........p-1}}` The number of A in `T_p`, such that A is either symmetric or skew-symmetric or both, and det (A) divisible by p is

Options

A. 2p-1
B. `p^(3)-5p`
C. 3p-4
D. `p^(3)-p^(2)`

Answer

To solve the problem step by step, we will analyze the conditions for the matrices in the set \( T_p \) and count the valid matrices that are either symmetric or skew-symmetric, or both, and have a determinant divisible by \( p \).

### Step 1: Define the matrix and its determinant
The matrices in the set \( T_p \) are of the form:
\[ A = \begin{pmatrix} a & b \\ c & a \end{pmatrix} \]
where \( a, b, c \) are elements from the set \( \{0, 1, 2, \ldots, p-1\} \).

The determinant of matrix \( A \) is given by:
\[ \text{det}(A) = a^2 - bc \]
We need to find the cases where \( \text{det}(A) \) is divisible by \( p \).

### Step 2: Analyze symmetric matrices
For a matrix to be symmetric, we require \( b = c \). Thus, the determinant becomes:
\[ \text{det}(A) = a^2 - b^2 = (a + b)(a - b) \]
For this determinant to be divisible by \( p \), either \( a + b \equiv 0 \mod p \) or \( a - b \equiv 0 \mod p \).

- **Case 1:** \( a + b \equiv 0 \mod p \)
This implies \( b = p - a \). The values of \( a \) can range from \( 0 \) to \( p-1 \), leading to \( p \) valid pairs \( (a, b) \).

- **Case 2:** \( a - b \equiv 0 \mod p \)
This implies \( a = b \). The values of \( a \) can also range from \( 0 \) to \( p-1 \), leading to \( p \) valid pairs \( (a, b) \).

However, we have counted the case \( b = c = 0 \) twice (once in each case). Therefore, we need to adjust for this double counting.

### Step 3: Count symmetric matrices
The total number of symmetric matrices where \( \text{det}(A) \equiv 0 \mod p \) is:
\[ p + (p - 1) = 2p - 1 \]
This accounts for all symmetric cases.

### Step 4: Analyze skew-symmetric matrices
For a matrix to be skew-symmetric, we require \( b = -c \). However, since \( b \) and \( c \) must be in the range \( \{0, 1, 2, \ldots, p-1\} \), the only possibility is \( b = c = 0 \).

In this case, the determinant becomes:
\[ \text{det}(A) = a^2 \]
This is divisible by \( p \) when \( a = 0 \). Thus, there is only **one valid skew-symmetric matrix**:
\[ A = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

### Step 5: Combine counts
The total number of matrices that are either symmetric or skew-symmetric or both is:
\[ (2p - 1) + 1 = 2p - 1 + 1 = 2p \]

### Final Answer
Thus, the number of matrices \( A \) in \( T_p \) such that \( A \) is either symmetric or skew-symmetric or both, and \( \text{det}(A) \) is divisible by \( p \) is:
\[ \boxed{2p - 1} \]