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JEE Advance - Mathematics (2010 - Paper 1 Offline - No. 26)

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

$$ \mathrm{T}_{\mathrm{p}}=\left\{\mathrm{A}=\left[\begin{array}{ll} a & b \\ c & a \end{array}\right]: a, b, c \in\{0,1,2, \ldots, p-1\}\right\} $$

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

$$ \mathrm{T}_{\mathrm{p}}=\left\{\mathrm{A}=\left[\begin{array}{ll} a & b \\ c & a \end{array}\right]: a, b, c \in\{0,1,2, \ldots, p-1\}\right\} $$

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

$$ \mathrm{T}_{\mathrm{p}}=\left\{\mathrm{A}=\left[\begin{array}{ll} a & b \\ c & a \end{array}\right]: a, b, c \in\{0,1,2, \ldots, p-1\}\right\} $$

The number of $A$ in $T_p$ such that $A$ is either symmetric or skew-symmetric or both, and $\operatorname{det}(\mathrm{A}) \operatorname{divisible}$ by $p$ is :
$(p-1)^2$
$2(p-1)$
$(p-1)^2+1$
$2 p-1$

Explanation

We must have $a^2-b^2=1 < p$

$$ (a+b)(a-b)=1 < p $$

Either $a-b=0$ or $a+b$ is a multiple of $p$ when $a=b$ number of matrices is $p$ and when $a+b$ $=$ multiple of $p$.

$$ \Rightarrow a, b \text { has } p-1 $$

$$ \begin{aligned} \text { Total number of matrices } & =p+p-1 \\\\ & =2 p-1 \end{aligned} $$

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