Step-by-step Derivations

Let $S$ be the set of symmetric matrices in $T_p$, $K$ be the set of skew-symmetric matrices in $T_p$, and $D$ be the set of matrices in $T_p$ whose determinant is divisible by $p$. We are asked to find the number of matrices $A$ such that $A \in (S \cup K)$ and $A \in D$. This is the size of the set $(S \cup K) \cap D$.

Using the distributive property of set intersection over union, we have: $(S \cup K) \cap D = (S \cap D) \cup (K \cap D)$ By the Principle of Inclusion-Exclusion, the size of this set is: $|(S \cap D) \cup (K \cap D)| = |S \cap D| + |K \cap D| - |S \cap K \cap D|$

We will now calculate the size of each of these three sets.

1. Counting Symmetric Matrices with Determinant Divisible by p ($|S \cap D|$)

A matrix $A = \begin{bmatrix} a & b \\ c & a \end{bmatrix}$ is symmetric if $A^T = A$. This means $\begin{bmatrix} a & c \\ b & a \end{bmatrix} = \begin{bmatrix} a & b \\ c & a \end{bmatrix}$, which implies $b=c$. So, a symmetric matrix in $T_p$ has the form $A = \begin{bmatrix} a & b \\ b & a \end{bmatrix}$.

The determinant of this matrix is $\det(A) = a^2 - b^2$. The condition that the determinant is divisible by $p$ means $\det(A) \equiv 0 \pmod{p}$. $a^2 - b^2 \equiv 0 \pmod{p}$ $(a-b)(a+b) \equiv 0 \pmod{p}$ Since $p$ is a prime number, this implies that either $(a-b) \equiv 0 \pmod{p}$ or $(a+b) \equiv 0 \pmod{p}$. That is, $a \equiv b \pmod{p}$ or $a \equiv -b \pmod{p}$.

We need to count the number of pairs $(a, b)$ with $a, b \in \{0, 1, \dots, p-1\}$ that satisfy this condition.

• Case 1: $a=b$ For any choice of $a$ from the $p$ possibilities, $b$ is fixed. The pairs are $(0,0), (1,1), \dots, (p-1, p-1)$. There are $p$ such pairs.

• Case 2: $a=-b \pmod{p}$ (or $a+b=p$ or $a+b=0$) For any choice of $a$ from the $p$ possibilities, $b$ is uniquely determined as $b = (-a) \pmod{p}$. For example, if $a=1, b=p-1$. If $a=0, b=0$. There are $p$ such pairs.

• Intersection of cases: We need to find pairs that satisfy both $a=b$ and $a=-b$. This means $a=-a \implies 2a \equiv 0 \pmod{p}$. Since $p$ is an odd prime, $\gcd(2, p) = 1$, so this implies $a=0$. If $a=0$, then $b=0$. The only pair in the intersection is $(0,0)$.

Using the Principle of Inclusion-Exclusion for counting the pairs $(a,b)$: Number of pairs = (pairs with $a=b$) + (pairs with $a=-b$) - (pairs in intersection) Number of pairs = $p + p - 1 = 2p-1$. For each such pair $(a,b)$, the matrix is determined since $c=b$. Thus, $|S \cap D| = 2p-1$.

2. Counting Skew-Symmetric Matrices with Determinant Divisible by p ($|K \cap D|$)

A matrix $A = \begin{bmatrix} a & b \\ c & a \end{bmatrix}$ is skew-symmetric if $A^T = -A$. This means $\begin{bmatrix} a & c \\ b & a \end{bmatrix} = \begin{bmatrix} -a & -b \\ -c & -a \end{bmatrix}$. This gives the conditions:

• $a = -a \implies 2a = 0$. Since $p$ is an odd prime, this implies $a=0$.
• $c = -b$. So, a skew-symmetric matrix in $T_p$ must be of the form $A = \begin{bmatrix} 0 & b \\ -b & 0 \end{bmatrix}$.

The determinant is $\det(A) = 0^2 - b(-b) = b^2$. The condition $\det(A)$ is divisible by $p$ means $b^2 \equiv 0 \pmod{p}$. Since $p$ is prime, this implies $b \equiv 0 \pmod{p}$, so $b=0$. If $b=0$, then $c=-b=0$. We also have $a=0$. The only matrix satisfying these conditions is the zero matrix $A = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$. Thus, $|K \cap D| = 1$.

3. Counting Matrices that are Both Symmetric and Skew-Symmetric with Determinant Divisible by p ($|S \cap K \cap D|$)

A matrix that is both symmetric and skew-symmetric must satisfy the conditions from both cases. From symmetric: $b=c$. From skew-symmetric: $a=0$ and $c=-b$. Substituting $b=c$ into $c=-b$ gives $b=-b \implies 2b=0$. Since $p$ is an odd prime, $b=0$. Then $c=0$. And $a=0$. This is the zero matrix. The determinant of the zero matrix is 0, which is divisible by $p$. So the zero matrix is in this set. Thus, $|S \cap K \cap D| = 1$.

Final Calculation

The total number of matrices is: $|S \cap D| + |K \cap D| - |S \cap K \cap D| = (2p-1) + 1 - 1 = 2p-1$

Conclusion

The number of matrices $A$ in $T_p$ that are either symmetric or skew-symmetric or both, and have a determinant divisible by $p$, is $2p-1$. This corresponds to option D.