IIT JAM Exam > IIT JAM Questions > Which of the following statement is correct?a...
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Which of the following statement is correct?
- a)The matrix A and its transpose AT have the same characteristic polynomial.
- b)Similar matrices have not the same characteristic polynomial.
- c)The trace of two similar matrices are not same.
- d)The determinant of two similar matrices is not same.
Correct answer is option 'A'. Can you explain this answer?
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Which of the following statement is correct?a)The matrix A and its tra...
Explanation:
Characteristic polynomial:
The characteristic polynomial of a square matrix A is defined as the polynomial det(A - λI) where det denotes the determinant, I is the identity matrix, and λ is a scalar variable.
Statement (a):
The matrix A and its transpose AT have the same characteristic polynomial.
This statement is true. The proof is given below:
- Let A be a square matrix of order n.
- Then, the characteristic polynomial of A is det(A - λI) where I is the n × n identity matrix.
- Now, consider the transpose of A, denoted by AT.
- The characteristic polynomial of AT is det(AT - λI).
- We know that (AT - λI) = (A - λI)T.
- Therefore, det(AT - λI) = det((A - λI)T) = det(A - λI).
- Hence, the matrix A and its transpose AT have the same characteristic polynomial.
Statement (b):
Similar matrices have not the same characteristic polynomial.
This statement is false. Similar matrices have the same characteristic polynomial.
- Two matrices A and B are said to be similar if there exists an invertible matrix P such that B = P-1AP.
- If A and B are similar matrices, then they have the same eigenvalues.
- The eigenvalues of a matrix are the roots of its characteristic polynomial.
- Hence, similar matrices have the same characteristic polynomial.
Statement (c):
The trace of two similar matrices are not same.
This statement is false. Similar matrices have the same trace.
- The trace of a matrix is the sum of its diagonal elements.
- If A and B are similar matrices, then they have the same characteristic polynomial and hence the same eigenvalues.
- The sum of the eigenvalues of a matrix is equal to its trace.
- Therefore, similar matrices have the same trace.
Statement (d):
The determinant of two similar matrices is not same.
This statement is false. Similar matrices have the same determinant.
- If A and B are similar matrices, then they have the same characteristic polynomial and hence the same eigenvalues.
- The determinant of a matrix is equal to the product of its eigenvalues.
- Therefore, similar matrices have the same determinant.
Characteristic polynomial:
The characteristic polynomial of a square matrix A is defined as the polynomial det(A - λI) where det denotes the determinant, I is the identity matrix, and λ is a scalar variable.
Statement (a):
The matrix A and its transpose AT have the same characteristic polynomial.
This statement is true. The proof is given below:
- Let A be a square matrix of order n.
- Then, the characteristic polynomial of A is det(A - λI) where I is the n × n identity matrix.
- Now, consider the transpose of A, denoted by AT.
- The characteristic polynomial of AT is det(AT - λI).
- We know that (AT - λI) = (A - λI)T.
- Therefore, det(AT - λI) = det((A - λI)T) = det(A - λI).
- Hence, the matrix A and its transpose AT have the same characteristic polynomial.
Statement (b):
Similar matrices have not the same characteristic polynomial.
This statement is false. Similar matrices have the same characteristic polynomial.
- Two matrices A and B are said to be similar if there exists an invertible matrix P such that B = P-1AP.
- If A and B are similar matrices, then they have the same eigenvalues.
- The eigenvalues of a matrix are the roots of its characteristic polynomial.
- Hence, similar matrices have the same characteristic polynomial.
Statement (c):
The trace of two similar matrices are not same.
This statement is false. Similar matrices have the same trace.
- The trace of a matrix is the sum of its diagonal elements.
- If A and B are similar matrices, then they have the same characteristic polynomial and hence the same eigenvalues.
- The sum of the eigenvalues of a matrix is equal to its trace.
- Therefore, similar matrices have the same trace.
Statement (d):
The determinant of two similar matrices is not same.
This statement is false. Similar matrices have the same determinant.
- If A and B are similar matrices, then they have the same characteristic polynomial and hence the same eigenvalues.
- The determinant of a matrix is equal to the product of its eigenvalues.
- Therefore, similar matrices have the same determinant.
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Which of the following statement is correct?a)The matrix A and its transpose AT have the same characteristic polynomial.b)Similar matrices have not the same characteristic polynomial.c)The trace of two similar matrices are not same.d)The determinant of two similar matrices is not same.Correct answer is option 'A'. Can you explain this answer?
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Which of the following statement is correct?a)The matrix A and its transpose AT have the same characteristic polynomial.b)Similar matrices have not the same characteristic polynomial.c)The trace of two similar matrices are not same.d)The determinant of two similar matrices is not same.Correct answer is option 'A'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Which of the following statement is correct?a)The matrix A and its transpose AT have the same characteristic polynomial.b)Similar matrices have not the same characteristic polynomial.c)The trace of two similar matrices are not same.d)The determinant of two similar matrices is not same.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statement is correct?a)The matrix A and its transpose AT have the same characteristic polynomial.b)Similar matrices have not the same characteristic polynomial.c)The trace of two similar matrices are not same.d)The determinant of two similar matrices is not same.Correct answer is option 'A'. Can you explain this answer?.
Which of the following statement is correct?a)The matrix A and its transpose AT have the same characteristic polynomial.b)Similar matrices have not the same characteristic polynomial.c)The trace of two similar matrices are not same.d)The determinant of two similar matrices is not same.Correct answer is option 'A'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Which of the following statement is correct?a)The matrix A and its transpose AT have the same characteristic polynomial.b)Similar matrices have not the same characteristic polynomial.c)The trace of two similar matrices are not same.d)The determinant of two similar matrices is not same.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statement is correct?a)The matrix A and its transpose AT have the same characteristic polynomial.b)Similar matrices have not the same characteristic polynomial.c)The trace of two similar matrices are not same.d)The determinant of two similar matrices is not same.Correct answer is option 'A'. Can you explain this answer?.
Solutions for Which of the following statement is correct?a)The matrix A and its transpose AT have the same characteristic polynomial.b)Similar matrices have not the same characteristic polynomial.c)The trace of two similar matrices are not same.d)The determinant of two similar matrices is not same.Correct answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for IIT JAM.
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Which of the following statement is correct?a)The matrix A and its transpose AT have the same characteristic polynomial.b)Similar matrices have not the same characteristic polynomial.c)The trace of two similar matrices are not same.d)The determinant of two similar matrices is not same.Correct answer is option 'A'. Can you explain this answer?, a detailed solution for Which of the following statement is correct?a)The matrix A and its transpose AT have the same characteristic polynomial.b)Similar matrices have not the same characteristic polynomial.c)The trace of two similar matrices are not same.d)The determinant of two similar matrices is not same.Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of Which of the following statement is correct?a)The matrix A and its transpose AT have the same characteristic polynomial.b)Similar matrices have not the same characteristic polynomial.c)The trace of two similar matrices are not same.d)The determinant of two similar matrices is not same.Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
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