IIT-JEE 2010 · Paper 1 · Mathematics Q42

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MathematicsSingle correct+3 / -1Question 42 of 84

Passage

Paragraph for Questions 42 to 44

Let p be an odd prime number and $T_p$ be the following set of $2\times2$ matrices:
$T_p=\left\{A=\begin{bmatrix}a&b\\c&a\end{bmatrix}:a,b,c\in\{0,1,2,\ldots,p-1\}\right\}$

The number of A in $T_p$ such that A is either symmetric or skew-symmetric or both, and det (A) divisible by p is

(p-1)^2

2(p-1)

(p-1)^2+1

2p-1

Answer and worked solution

Official answer: D

Symmetric means $b=c$ ; then $\det A=a^2-b^2=(a-b)(a+b)$ , divisible by the prime $p$ when $a\equiv b$ or $a\equiv-b\pmod p$ .

$a=b$ : $p$ matrices.
$a+b=p$ with $1\le a\le p-1$ : $p-1$ matrices (these have $a\ne b$ since $p$ is odd).

Skew-symmetric needs $a=0$ and $c=-b$ ; with entries in $\{0,\ldots,p-1\}$ only the zero matrix qualifies, already counted.

Total. $p+(p-1)=2p-1$ .

(D)

Topics:Special matrices Fundamental principle of counting

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IIT-JEE 2010 · Paper 1, Mathematics question 42