Electronics and Communication Engineering (ECE) Exam > Electronics and Communication Engineering (ECE) Questions > If 1 and 3 are the eigenvalues of a square ma...
Start Learning for Free
If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal to
- a)13 (A - I2 )
- b)13A - 12 I2
- c)12 (A - I2)
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal t...
Since 1 and 3 are the eigenvalues of A so the characteristic equation of A is
Also, by Cayley–Hamilton theorem, every square matrix satisfies its own characteristic equation so
Most Upvoted Answer
If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal t...
Solution:
Given, the eigenvalues of matrix A are 1 and 3.
We need to find A^3.
Let's find the matrix A first.
Let v be an eigenvector of A corresponding to eigenvalue λ.
Then, Av = λv.
Let's find the eigenvectors corresponding to eigenvalues 1 and 3.
For λ = 1,
Av = v
⇒ A - I = 0
⇒ A = I
For λ = 3,
Av = 3v
⇒ Av - 3v = 0
⇒ (A - 3I)v = 0
This implies that the null space of (A - 3I) is non-trivial and contains eigenvectors corresponding to the eigenvalue 3.
Let x be an eigenvector of A corresponding to the eigenvalue 3. Then,
Ax = 3x
⇒ A^2x = A(3x) = 3Ax = 9x
⇒ A^3x = A(9x) = 9Ax = 27x
So, A^3 has eigenvalue 27 corresponding to eigenvector x.
Now, let's find the matrix A^3.
A^3x = 27x
⇒ A^3x = 27(Ax)
⇒ A^3 = 27A
Substituting A = I for eigenvalue 1 and A = 3I for eigenvalue 3, we get
A^3 = 27A
⇒ A^3 = 27(I + 2A)
⇒ A^3 = 27I + 54A
⇒ A^3 = 13(2A) + 12I
Therefore, A^3 = 13A - 12I. Hence, option (B) is correct.
Given, the eigenvalues of matrix A are 1 and 3.
We need to find A^3.
Let's find the matrix A first.
Let v be an eigenvector of A corresponding to eigenvalue λ.
Then, Av = λv.
Let's find the eigenvectors corresponding to eigenvalues 1 and 3.
For λ = 1,
Av = v
⇒ A - I = 0
⇒ A = I
For λ = 3,
Av = 3v
⇒ Av - 3v = 0
⇒ (A - 3I)v = 0
This implies that the null space of (A - 3I) is non-trivial and contains eigenvectors corresponding to the eigenvalue 3.
Let x be an eigenvector of A corresponding to the eigenvalue 3. Then,
Ax = 3x
⇒ A^2x = A(3x) = 3Ax = 9x
⇒ A^3x = A(9x) = 9Ax = 27x
So, A^3 has eigenvalue 27 corresponding to eigenvector x.
Now, let's find the matrix A^3.
A^3x = 27x
⇒ A^3x = 27(Ax)
⇒ A^3 = 27A
Substituting A = I for eigenvalue 1 and A = 3I for eigenvalue 3, we get
A^3 = 27A
⇒ A^3 = 27(I + 2A)
⇒ A^3 = 27I + 54A
⇒ A^3 = 13(2A) + 12I
Therefore, A^3 = 13A - 12I. Hence, option (B) is correct.
Free Test
FREE
| Start Free Test |
Community Answer
If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal t...
Solution:
Given, the eigenvalues of a square matrix A are 1 and 3.
Eigenvalues of Matrix A
The eigenvalues of matrix A are the values λ for which Ax = λx has a non-zero solution x. Here, x is known as the eigenvector of matrix A.
Let x be an eigenvector of matrix A, then we have Ax = λx. Multiplying both sides by A, we get A2x = λAx = λ2x. Continuing this process, we get Anx = λnx, where n is a positive integer.
Power of Matrix A
If A is a square matrix and λ is its eigenvalue with eigenvector x, then A2, A3, …, An also have eigenvalue λ with eigenvector x. Therefore, we can say that λn is the eigenvalue of An with eigenvector x.
Given that the eigenvalues of matrix A are 1 and 3, we can say that A2 has eigenvalues 12 = 1 and 32 = 9.
Now, we need to find the value of A3.
Using the formula, A2x = λx, we can write Ax = λ1/2x.
Multiplying both sides by A, we get A2x = λA1/2x = λλ1/2x = λ3/2x.
Again, multiplying both sides by A, we get A3x = λA2x = λ(λ1/2x) = λ5/2x.
Therefore, the eigenvalues of A3 are 15/2 and 35/2.
Option (b) says that A3 = 13A - 12I2.
Let's verify this option.
A3 = 13A - 12I2
=> A3 = 13(A - I2) + I2
=> A3 = 13(1/2A) + 3/2I2
=> A3 = (13/2)A + (3/2)I2
The eigenvalues of (13/2)A + (3/2)I2 are (13/2) and (3/2).
Therefore, option (b) is the correct answer.
Hence, the answer is option (b) 13A - 12I2.
Given, the eigenvalues of a square matrix A are 1 and 3.
Eigenvalues of Matrix A
The eigenvalues of matrix A are the values λ for which Ax = λx has a non-zero solution x. Here, x is known as the eigenvector of matrix A.
Let x be an eigenvector of matrix A, then we have Ax = λx. Multiplying both sides by A, we get A2x = λAx = λ2x. Continuing this process, we get Anx = λnx, where n is a positive integer.
Power of Matrix A
If A is a square matrix and λ is its eigenvalue with eigenvector x, then A2, A3, …, An also have eigenvalue λ with eigenvector x. Therefore, we can say that λn is the eigenvalue of An with eigenvector x.
Given that the eigenvalues of matrix A are 1 and 3, we can say that A2 has eigenvalues 12 = 1 and 32 = 9.
Now, we need to find the value of A3.
Using the formula, A2x = λx, we can write Ax = λ1/2x.
Multiplying both sides by A, we get A2x = λA1/2x = λλ1/2x = λ3/2x.
Again, multiplying both sides by A, we get A3x = λA2x = λ(λ1/2x) = λ5/2x.
Therefore, the eigenvalues of A3 are 15/2 and 35/2.
Option (b) says that A3 = 13A - 12I2.
Let's verify this option.
A3 = 13A - 12I2
=> A3 = 13(A - I2) + I2
=> A3 = 13(1/2A) + 3/2I2
=> A3 = (13/2)A + (3/2)I2
The eigenvalues of (13/2)A + (3/2)I2 are (13/2) and (3/2).
Therefore, option (b) is the correct answer.
Hence, the answer is option (b) 13A - 12I2.
Attention Electronics and Communication Engineering (ECE) Students!
To make sure you are not studying endlessly, EduRev has designed Electronics and Communication Engineering (ECE) study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Electronics and Communication Engineering (ECE).
|
Explore Courses for Electronics and Communication Engineering (ECE) exam
|
|
Similar Electronics and Communication Engineering (ECE) Doubts
Top Courses for Electronics and Communication Engineering (ECE)View all
If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal toa)13(A -I2 )b)13A - 12I2c)12 (A - I2)d)None of theseCorrect answer is option 'B'. Can you explain this answer?
Question Description
If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal toa)13(A -I2 )b)13A - 12I2c)12 (A - I2)d)None of theseCorrect answer is option 'B'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal toa)13(A -I2 )b)13A - 12I2c)12 (A - I2)d)None of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal toa)13(A -I2 )b)13A - 12I2c)12 (A - I2)d)None of theseCorrect answer is option 'B'. Can you explain this answer?.
If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal toa)13(A -I2 )b)13A - 12I2c)12 (A - I2)d)None of theseCorrect answer is option 'B'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal toa)13(A -I2 )b)13A - 12I2c)12 (A - I2)d)None of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal toa)13(A -I2 )b)13A - 12I2c)12 (A - I2)d)None of theseCorrect answer is option 'B'. Can you explain this answer?.
Solutions for If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal toa)13(A -I2 )b)13A - 12I2c)12 (A - I2)d)None of theseCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Electronics and Communication Engineering (ECE).
Download more important topics, notes, lectures and mock test series for Electronics and Communication Engineering (ECE) Exam by signing up for free.
Here you can find the meaning of If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal toa)13(A -I2 )b)13A - 12I2c)12 (A - I2)d)None of theseCorrect answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal toa)13(A -I2 )b)13A - 12I2c)12 (A - I2)d)None of theseCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal toa)13(A -I2 )b)13A - 12I2c)12 (A - I2)d)None of theseCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal toa)13(A -I2 )b)13A - 12I2c)12 (A - I2)d)None of theseCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal toa)13(A -I2 )b)13A - 12I2c)12 (A - I2)d)None of theseCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice Electronics and Communication Engineering (ECE) tests.
|
|
Explore Courses for Electronics and Communication Engineering (ECE) 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.
x
![]()
For Your Perfect Score in Electronics and Communication Engineering (ECE)
The Best you need at One Place
|
© 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