SATHEE: Matrices and Determinants 1 Question 17

Matrices and Determinants 1 Question 17

Passage II

Let $p$ be an odd prime number and $T_{p}$ be the following set of $2 \times 2$ matrices

$T_{p} = A = [\begin{matrix} a & b \\ c & a \end{matrix}]; a, b, c \in 0, 1, 2, \dots, p - 1$

(2010)

17. The number of $A$ in $T_{p}$ such that $\det (A)$ is not divisible by $p$ , is

(a) $2 p^{2}$

(c) $p^{3} - 3 p$

(b) $p^{3} - 5 p$

(d) $p^{3} - p^{2}$

Show Answer

Answer:

Correct Answer: 17. ( d)

Solution:

The number of matrices for which $p$ does not divide $Tr (A) = (p - 1) p^{2}$ of these $(p - 1)^{2}$ are such that $p$ divides $| A |$ . The number of matrices for which $p$ divides $Tr (A)$ and $p$ does not divides $| A |$ are $(p - 1)^{2}$ .

$∴$ Required number $= (p - 1) p^{2} - (p - 1)^{2} + (p - 1)^{2}$

$= p^{3} - p^{2}$

Privacy Policy

Powered by Prutor@IITK

sathee@iitk.ac.in

Media coverage of SATHEE

Play Store

App Store

Ask SATHEE

Welcome to SATHEE !
Select from 'Menu' to explore our services, or ask SATHEE to get started. Let's embark on this journey of growth together! 🌐📚🚀🎓

I'm relatively new and can sometimes make mistakes.
If you notice any error, such as an incorrect solution, please use the thumbs down icon to aid my learning.
To begin your journey now, click on "I understand".

Matrices and Determinants 1 Question 17

17. The number of A in Tp such that det(A) is not divisible by p, is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

17. The number of $A$ in $T_{p}$ such that $\det (A)$ is not divisible by $p$ , is