Let p be an odd prime number and T p be the following set of 2 × 2 matrices: Tp= A=[a b c a]: a., b.c .∈ 0.1,2 . ldots, 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

Question

Let p be an odd prime number and T p be the following set of 2 × 2 matrices: Tp= A=[a b c a]: a., b.c .∈ 0.1,2 . ldots, 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 (A) (p-1)2 (B) 2(p-1) (C) (p-1)2+1 (D) 2 p -1

Tardigrade Admin · Accepted Answer

We must have a2-b2=k p ⇒ (a+b)(a-b)=k p ⇒ either a - b =0 or a + b is a multiple of p and when a + b = multiple of p ⇒ a , b has p -1 ∴ Total number of matrices = p + p -1 =2 p -1

Q. Let $p$ be an odd prime number and $T_{p}$ be the following set of $2 \times 2$ matrices :
$T_{p} = {A = [a c b a] : a$ , b.c $\in {0.1, 2. \dots, p - 1}}$
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

$(p - 1)^{2}$

$2 (p - 1)$

$(p - 1)^{2} + 1$

$2 p - 1$

We must have $a^{2} - b^{2} = k p$
$\Rightarrow (a + b) (a - b) = k p$
$\Rightarrow$ either $a - b = 0$ or $a + b$ is a multiple of $p$
and when $a + b =$ multiple of $p$
$\Rightarrow a, b$ has $p - 1$
$∴$ Total number of matrices $= p + p - 1$
$= 2 p - 1$

Q. Let p be an odd prime number and Tp​ be the following set of 2×2 matrices : Tp​={A=[ac​ba​]:a, b.c ∈{0.1,2.…,p−1}} The number of A in Tp​ such that A is either symmetric or skew-symmetric or both, and det (A) divisible by p is

(p−1)2

2(p−1)

(p−1)2+1

2p−1