IIT-JEE 2010 · Paper 1 · Mathematics Q43

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MathematicsSingle correct+3 / -1Question 43 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 the trace of A is not divisible by p but det (A) is divisible by p is

[Note: The trace of a matrix is the sum of its diagonal entries.]

(p-1)(p^2-p+1)

p^3-(p-1)^2

(p-1)^2

(p-1)(p^2-2)

Answer and worked solution

Official answer: C

Trace not divisible by $p$ . $\operatorname{tr}A=2a$ and $p$ is odd, so $a\ne0$ .

Determinant divisible by $p$ . $a^2-bc\equiv0$ , i.e. $bc\equiv a^2\not\equiv0$ . Then $b\ne0$ , and for each of the $p-1$ choices of $b$ there is exactly one $c$ (because $\mathbb Z_p$ is a field).

Count. $(p-1)$ choices of $a$ times $(p-1)$ pairs $(b,c)$ : $(p-1)^2$ .

(C)

Topics:Determinants and properties Fundamental principle of counting

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IIT-JEE 2010 · Paper 1, Mathematics question 43