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IIT JEE Advanced 2010 Paper 1

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Let p be an odd prime number and

T_p

be the following set of 2 × 2 matrices:

T_p

=

\{ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a , b , c \in \{0 , 1 , \dots , p - 1\} \}

Section: Mathematics

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Question : 74 of 84

Marks: +1, -0

The number of A in

T_p

such that det (A) is not divisible by p is

[JEE Adv 2010 P1]

$2p^2$
$p^3 − 5p$
$p^3 − 3p$
$p^3 − p^2$

Solution:

The total number of A = p × p × p =

p^3

.

Number of A such that det(A) is divisible by p is

(p-1)^2

+

=

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

=

p^2

The required number is

p^3-p^2

.

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