Mathematics Exam > Mathematics Questions > Let H denote the group of all 2*2 invertible ...
Start Learning for Free
Let H denote the group of all 2*2 invertible matrices over Z5 under matrix multiplication, then the order of matrix [2 3;1 2]in H is (a) 1 (b) 2 (c) 3 (d) 4?
Most Upvoted Answer
Let H denote the group of all 2*2 invertible matrices over Z5 under ma...
Group H and Matrix [2 3; 1 2]
In this question, we are given a group H, which consists of all 2x2 invertible matrices over Z5 (the integers modulo 5) under matrix multiplication. We need to find the order of the matrix [2 3; 1 2] in this group.
Definition of Order
The order of an element in a group is the smallest positive integer n such that the element raised to the power n gives the identity element of the group. In this case, the identity element is the 2x2 identity matrix [1 0; 0 1].
Calculating Powers of the Matrix
To find the order of the matrix [2 3; 1 2], we need to calculate its powers until we reach the identity matrix.
Power 1:
[2 3; 1 2]
Power 2:
[2 3; 1 2] * [2 3; 1 2] = [7 12; 4 7] = [2 2; 4 2] (using modulo 5 arithmetic)
Power 3:
[2 2; 4 2] * [2 3; 1 2] = [6 10; 10 16] = [1 0; 0 1] (using modulo 5 arithmetic)
Order of the Matrix
We can see that the matrix [2 3; 1 2] raised to the power 3 gives the identity matrix [1 0; 0 1]. Therefore, the order of the matrix [2 3; 1 2] in the group H is 3.
Answer: (c) 3
The matrix [2 3; 1 2] has an order of 3 in the group H.
In this question, we are given a group H, which consists of all 2x2 invertible matrices over Z5 (the integers modulo 5) under matrix multiplication. We need to find the order of the matrix [2 3; 1 2] in this group.
Definition of Order
The order of an element in a group is the smallest positive integer n such that the element raised to the power n gives the identity element of the group. In this case, the identity element is the 2x2 identity matrix [1 0; 0 1].
Calculating Powers of the Matrix
To find the order of the matrix [2 3; 1 2], we need to calculate its powers until we reach the identity matrix.
Power 1:
[2 3; 1 2]
Power 2:
[2 3; 1 2] * [2 3; 1 2] = [7 12; 4 7] = [2 2; 4 2] (using modulo 5 arithmetic)
Power 3:
[2 2; 4 2] * [2 3; 1 2] = [6 10; 10 16] = [1 0; 0 1] (using modulo 5 arithmetic)
Order of the Matrix
We can see that the matrix [2 3; 1 2] raised to the power 3 gives the identity matrix [1 0; 0 1]. Therefore, the order of the matrix [2 3; 1 2] in the group H is 3.
Answer: (c) 3
The matrix [2 3; 1 2] has an order of 3 in the group H.
|
Explore Courses for Mathematics exam
|
|
Similar Mathematics Doubts
Let H denote the group of all 2*2 invertible matrices over Z5 under matrix multiplication, then the order of matrix [2 3;1 2]in H is (a) 1 (b) 2 (c) 3 (d) 4?
Question Description
Let H denote the group of all 2*2 invertible matrices over Z5 under matrix multiplication, then the order of matrix [2 3;1 2]in H is (a) 1 (b) 2 (c) 3 (d) 4? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let H denote the group of all 2*2 invertible matrices over Z5 under matrix multiplication, then the order of matrix [2 3;1 2]in H is (a) 1 (b) 2 (c) 3 (d) 4? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let H denote the group of all 2*2 invertible matrices over Z5 under matrix multiplication, then the order of matrix [2 3;1 2]in H is (a) 1 (b) 2 (c) 3 (d) 4?.
Let H denote the group of all 2*2 invertible matrices over Z5 under matrix multiplication, then the order of matrix [2 3;1 2]in H is (a) 1 (b) 2 (c) 3 (d) 4? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let H denote the group of all 2*2 invertible matrices over Z5 under matrix multiplication, then the order of matrix [2 3;1 2]in H is (a) 1 (b) 2 (c) 3 (d) 4? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let H denote the group of all 2*2 invertible matrices over Z5 under matrix multiplication, then the order of matrix [2 3;1 2]in H is (a) 1 (b) 2 (c) 3 (d) 4?.
Solutions for Let H denote the group of all 2*2 invertible matrices over Z5 under matrix multiplication, then the order of matrix [2 3;1 2]in H is (a) 1 (b) 2 (c) 3 (d) 4? in English & in Hindi are available as part of our courses for Mathematics.
Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free.
Here you can find the meaning of Let H denote the group of all 2*2 invertible matrices over Z5 under matrix multiplication, then the order of matrix [2 3;1 2]in H is (a) 1 (b) 2 (c) 3 (d) 4? defined & explained in the simplest way possible. Besides giving the explanation of
Let H denote the group of all 2*2 invertible matrices over Z5 under matrix multiplication, then the order of matrix [2 3;1 2]in H is (a) 1 (b) 2 (c) 3 (d) 4?, a detailed solution for Let H denote the group of all 2*2 invertible matrices over Z5 under matrix multiplication, then the order of matrix [2 3;1 2]in H is (a) 1 (b) 2 (c) 3 (d) 4? has been provided alongside types of Let H denote the group of all 2*2 invertible matrices over Z5 under matrix multiplication, then the order of matrix [2 3;1 2]in H is (a) 1 (b) 2 (c) 3 (d) 4? theory, EduRev gives you an
ample number of questions to practice Let H denote the group of all 2*2 invertible matrices over Z5 under matrix multiplication, then the order of matrix [2 3;1 2]in H is (a) 1 (b) 2 (c) 3 (d) 4? tests, examples and also practice Mathematics tests.
|
|
Explore Courses for Mathematics exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup