Mathematics Exam > Mathematics Questions > Let the characteristic equation of a matrix (...
Start Learning for Free
Let the characteristic equation of a matrix (x - α)3 + (x - β)3 a be λ2 - λ - 1 then
- a)M-1 does not exists
- b)M-1 exists but cannot be determined from the data
- c)M-1 = M - I
- d)M-1 = M + i
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Let the characteristic equation of a matrix (x - α)3 + (x - &bet...
Understanding the Characteristic Equation
The characteristic equation given is λ^2 - λ - 1. To analyze the implications of this equation, we first need to find its roots using the quadratic formula. The roots will help us determine the eigenvalues of the matrix M.
Eigenvalues and Matrix Inversion
The roots of the equation are:
- λ = (1 ± √5) / 2
These eigenvalues indicate whether the matrix M is invertible. A matrix is invertible if and only if zero is not an eigenvalue. Since neither root is zero, M is indeed invertible.
Matrix Inversion Properties
For a matrix M, the inverse can sometimes be expressed in terms of its eigenvalues. In this case, we have the condition that:
- M - I = 0
This expression suggests that the matrix M can be adjusted by subtracting the identity matrix, leading to the conclusion that:
- M - I is a matrix that can yield the identity when manipulated correctly.
Conclusion: Why Option C is Correct
Given the options, we find that:
- Option C states M^(-1) = M - I, which aligns with our understanding that subtracting the identity matrix brings us back to a form that is manageable and consistent with the eigenvalues we derived.
Thus, the correct answer is indeed:
- M^(-1) = M - I
This result highlights the unique relationship between M and its inverse in this specific characteristic equation context.
The characteristic equation given is λ^2 - λ - 1. To analyze the implications of this equation, we first need to find its roots using the quadratic formula. The roots will help us determine the eigenvalues of the matrix M.
Eigenvalues and Matrix Inversion
The roots of the equation are:
- λ = (1 ± √5) / 2
These eigenvalues indicate whether the matrix M is invertible. A matrix is invertible if and only if zero is not an eigenvalue. Since neither root is zero, M is indeed invertible.
Matrix Inversion Properties
For a matrix M, the inverse can sometimes be expressed in terms of its eigenvalues. In this case, we have the condition that:
- M - I = 0
This expression suggests that the matrix M can be adjusted by subtracting the identity matrix, leading to the conclusion that:
- M - I is a matrix that can yield the identity when manipulated correctly.
Conclusion: Why Option C is Correct
Given the options, we find that:
- Option C states M^(-1) = M - I, which aligns with our understanding that subtracting the identity matrix brings us back to a form that is manageable and consistent with the eigenvalues we derived.
Thus, the correct answer is indeed:
- M^(-1) = M - I
This result highlights the unique relationship between M and its inverse in this specific characteristic equation context.
|
Explore Courses for Mathematics exam
|
|
Similar Mathematics Doubts
Question Description
Let the characteristic equation of a matrix (x - α)3 + (x - β)3 a be λ2 - λ - 1 thena)M-1 does not existsb)M-1 exists but cannot be determined from the datac)M-1 = M - Id)M-1 = M + iCorrect answer is option 'C'. Can you explain this answer? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let the characteristic equation of a matrix (x - α)3 + (x - β)3 a be λ2 - λ - 1 thena)M-1 does not existsb)M-1 exists but cannot be determined from the datac)M-1 = M - Id)M-1 = M + iCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let the characteristic equation of a matrix (x - α)3 + (x - β)3 a be λ2 - λ - 1 thena)M-1 does not existsb)M-1 exists but cannot be determined from the datac)M-1 = M - Id)M-1 = M + iCorrect answer is option 'C'. Can you explain this answer?.
Let the characteristic equation of a matrix (x - α)3 + (x - β)3 a be λ2 - λ - 1 thena)M-1 does not existsb)M-1 exists but cannot be determined from the datac)M-1 = M - Id)M-1 = M + iCorrect answer is option 'C'. Can you explain this answer? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let the characteristic equation of a matrix (x - α)3 + (x - β)3 a be λ2 - λ - 1 thena)M-1 does not existsb)M-1 exists but cannot be determined from the datac)M-1 = M - Id)M-1 = M + iCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let the characteristic equation of a matrix (x - α)3 + (x - β)3 a be λ2 - λ - 1 thena)M-1 does not existsb)M-1 exists but cannot be determined from the datac)M-1 = M - Id)M-1 = M + iCorrect answer is option 'C'. Can you explain this answer?.
Solutions for Let the characteristic equation of a matrix (x - α)3 + (x - β)3 a be λ2 - λ - 1 thena)M-1 does not existsb)M-1 exists but cannot be determined from the datac)M-1 = M - Id)M-1 = M + iCorrect answer is option 'C'. Can you explain this answer? 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 the characteristic equation of a matrix (x - α)3 + (x - β)3 a be λ2 - λ - 1 thena)M-1 does not existsb)M-1 exists but cannot be determined from the datac)M-1 = M - Id)M-1 = M + iCorrect answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Let the characteristic equation of a matrix (x - α)3 + (x - β)3 a be λ2 - λ - 1 thena)M-1 does not existsb)M-1 exists but cannot be determined from the datac)M-1 = M - Id)M-1 = M + iCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for Let the characteristic equation of a matrix (x - α)3 + (x - β)3 a be λ2 - λ - 1 thena)M-1 does not existsb)M-1 exists but cannot be determined from the datac)M-1 = M - Id)M-1 = M + iCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of Let the characteristic equation of a matrix (x - α)3 + (x - β)3 a be λ2 - λ - 1 thena)M-1 does not existsb)M-1 exists but cannot be determined from the datac)M-1 = M - Id)M-1 = M + iCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Let the characteristic equation of a matrix (x - α)3 + (x - β)3 a be λ2 - λ - 1 thena)M-1 does not existsb)M-1 exists but cannot be determined from the datac)M-1 = M - Id)M-1 = M + iCorrect answer is option 'C'. Can you explain this answer? tests, examples and also practice Mathematics tests.
|
|
Explore Courses for Mathematics exam
|
|
Signup to solve all Doubts
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup