The number of A in T p such that A is either symmetric or skew-symmetric or both, and det ( A ) is divisible by p , is [2010]

Question

Let $p$ be an odd prime number and $T_{p}$ be the following set of $2 \times 2$ matrices:

$T_{p} = {A = [\begin{matrix} a & b \\ c & a \end{matrix}] : a, b, c \in {0, 1, 2, \dots, p - 1}}$ [2010]

Q. The number of $A$ in $T_{p}$ such that $A$ is either symmetric or skew-symmetric or both, and $\det (A)$ is divisible by $p$ , is

Smart Achievers · Accepted Answer

(4)

$Given, A = [\begin{matrix} a & b \\ c & a \end{matrix}], a, b, c \in {0, 1, 2, \dots, p - 1}$

If $A$ is a skew-symmetric matrix, then $a = 0,$ $b = - c$

$∴$ $| A |$ $= - b^{2}$

Thus, $p$ divides $| A |$ , only when $b = 0 .$ $\dots (i)$

Again, if $A$ is a symmetric matrix, then $b = c$ and $| A |$ $= a^{2} - b^{2}$

Thus, $p$ divides $| A |$ , if either $p$ divides $(a - b)$ or $p$ divides $(a + b) .$

$p$ divides $(a - b)$ , only when $a = b,$

i.e., $a = b \in {0, 1, 2, \dots, (p - 1)},$

i.e., $p$ choices. $\dots (i i)$

$p$ divides $(a + b)$ .

$\Rightarrow p$ choices, including $a = b = 0$ included in Eq. (i).

$∴$ Total number of choices are $(p + p - 1) = 2 p - 1$

Want Us to Email you About Special Offers & Updates?