Test Index

Matrices and Determinants

Show Para Hide Para

Paragraph for next 3 Questions

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

© examsnet.com

Question : 51 of 56

Marks: +1, -0

The number of A in

T_p

such that the trace of A is not divisible by p bu t det (A) is divisible by p is

(Note: The trace of a matrix is the sum of its diagonal entries.)

[JEE Adv 2010 P1]

(p − 1) ( $p^2$ − p + 1)
$p^3 − (p − 1)^2$
$(p − 1)^2$
$(p − 1) (p^2 − 2)$

Solution:

As Tr(A) is not divisible by p ⇒ a ≠ 0.

Now, det(A) is divisible by p ⇒

a^2

- bc is divisible by p.

Number of ways of selection of a, b, and c is

(p - 1)[(p - 1) × 1] =

(p -1)^2

© examsnet.com

Go to Question:

Prev Question

Next Question