Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices: $T_p = \left\{ A = \begin{bmatrix} a & b \\ c & a \end{bmatrix} : a, b, c \in \{0, 1, 2, \dots, p-1\} \right\}$. 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 -
Step-by-Step Solution
Key Concept: A matrix is symmetric if b=c, skew-symmetric if b=-c (mod p), and we need det(A) = a² - bc ≡ 0 (mod p). We count matrices satisfying these conditions separately and use inclusion-exclusion.
<p><strong>Step 1: Identify conditions.</strong> Matrix A is symmetric if b=c, skew-symmetric if c=-b (mod p), and we need det(A) = a² - bc ≡ 0 (mod p).</p><p><strong>Step 2: Count symmetric matrices with det(A) ≡ 0 (mod p).</strong> For symmetric: A = [a b; b a]. We need a² - b² ≡ 0 (mod p), so (a-b)(a+b) ≡ 0 (mod p). Since p is prime: either a ≡ b (mod p) or a ≡ -b (mod p).</p><p>Case 2a: a = b. Choose a ∈ {0,1,...,p-1} (p choices), b = a (determined). Count: p matrices.</p><p>Case 2b: a = -b (mod p), a ≠ b. For a,b ∈ {0,1,...,p-1}: if a = -b mod p, then b = p-a. When a=0, b=0 (already in Case 2a). For a ∈ {1,...,p-1}, we get b = p-a ∈ {1,...,p-1}, a ≠ b. Count: p-1 matrices.</p><p>Total symmetric: p + (p-1) = 2p-1.</p><p><strong>Step 3: Count skew-symmetric matrices with det(A) ≡ 0 (mod p).</strong> For skew-symmetric: A = [a b; -b a]. We need a² + b² ≡ 0 (mod p), so a² ≡ -b² (mod p).</p><p>For odd prime p: a² ≡ -b² (mod p) means a=b=0 (since -1 is not always a quadratic residue, and we'd need a²+b²≡0). When a=b=0, det = 0. Count: 1 matrix.</p><p><strong>Step 4: Apply inclusion-exclusion.</strong> A matrix both symmetric and skew-symmetric requires b=c AND c=-b, so 2b≡0 (mod p). For odd p: b=0, and c=0. Then A = [a 0; 0 a], with det = a² ≡ 0 (mod p), giving a=0. Only one matrix: [0 0; 0 0].</p><p>Total = (symmetric count) + (skew-symmetric count) - (both) = (2p-1) + 1 - 1 = 2p-1.</p><p><strong>∴ Answer: D</strong></p>
Correct Answer: D