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Let N be a nilpotent matrix of order 4 with real entries. Then which one of the following statements is true about eigenvalues of N?
- a)All eigenvalues are non zero real numbers.
- b)All eigenvalues are purely imaginary.
- c)Zero is the only eigenvalue.
- d)At least one eigenvalue is real and at least one eigenvalue has non zero imaginary part.
Correct answer is option 'C'. Can you explain this answer?
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Let N be a nilpotent matrix of order 4 with real entries. Then which o...
Let N be a nilpotent matrix of order 4.
⇒ N4 = 0; N satisfied its characteristic equation
i.e λ4 = 0
λ = 0 eigenvalue of N
Since, eigenvalue of N4 is λ4 where λ is eigenvalue of N.
zero is the only eigenvalue as eigenvalue of N is all zero.
⇒ N4 = 0; N satisfied its characteristic equation
i.e λ4 = 0
λ = 0 eigenvalue of N
Since, eigenvalue of N4 is λ4 where λ is eigenvalue of N.
zero is the only eigenvalue as eigenvalue of N is all zero.
Most Upvoted Answer
Let N be a nilpotent matrix of order 4 with real entries. Then which o...
Explanation:
To determine the eigenvalues of a nilpotent matrix N, we need to understand the properties of nilpotent matrices.
Definition:
A matrix N is called nilpotent if there exists a positive integer k such that N^k = 0, where 0 is the zero matrix.
In other words, a nilpotent matrix N is a square matrix for which there exists a positive integer k such that N raised to the power of k is the zero matrix.
Now let's analyze the options one by one:
a) All eigenvalues are non-zero real numbers:
This statement is not true for nilpotent matrices. Since N is nilpotent, there exists a positive integer k such that N^k = 0. If all eigenvalues were non-zero real numbers, then the matrix N^k would have non-zero eigenvalues, which contradicts the fact that N^k is the zero matrix.
b) All eigenvalues are purely imaginary:
This statement is not true for nilpotent matrices either. Similar to the previous case, if all eigenvalues were purely imaginary, then N^k would have purely imaginary eigenvalues, which contradicts the fact that N^k is the zero matrix.
c) Zero is the only eigenvalue:
This statement is true for nilpotent matrices. Since N is nilpotent, there exists a positive integer k such that N^k = 0. This means that the characteristic polynomial of N, which is det(N - λI), must have λ = 0 as a root. Therefore, zero is the only eigenvalue of N.
d) At least one eigenvalue is real and at least one eigenvalue has a non-zero imaginary part:
This statement is also not true for nilpotent matrices. As mentioned earlier, if all eigenvalues were real or had a non-zero imaginary part, then N^k would have eigenvalues that do not satisfy these conditions, which contradicts the fact that N^k is the zero matrix.
Therefore, the correct answer is option 'c'. Zero is the only eigenvalue of a nilpotent matrix of order 4 with real entries.
To determine the eigenvalues of a nilpotent matrix N, we need to understand the properties of nilpotent matrices.
Definition:
A matrix N is called nilpotent if there exists a positive integer k such that N^k = 0, where 0 is the zero matrix.
In other words, a nilpotent matrix N is a square matrix for which there exists a positive integer k such that N raised to the power of k is the zero matrix.
Now let's analyze the options one by one:
a) All eigenvalues are non-zero real numbers:
This statement is not true for nilpotent matrices. Since N is nilpotent, there exists a positive integer k such that N^k = 0. If all eigenvalues were non-zero real numbers, then the matrix N^k would have non-zero eigenvalues, which contradicts the fact that N^k is the zero matrix.
b) All eigenvalues are purely imaginary:
This statement is not true for nilpotent matrices either. Similar to the previous case, if all eigenvalues were purely imaginary, then N^k would have purely imaginary eigenvalues, which contradicts the fact that N^k is the zero matrix.
c) Zero is the only eigenvalue:
This statement is true for nilpotent matrices. Since N is nilpotent, there exists a positive integer k such that N^k = 0. This means that the characteristic polynomial of N, which is det(N - λI), must have λ = 0 as a root. Therefore, zero is the only eigenvalue of N.
d) At least one eigenvalue is real and at least one eigenvalue has a non-zero imaginary part:
This statement is also not true for nilpotent matrices. As mentioned earlier, if all eigenvalues were real or had a non-zero imaginary part, then N^k would have eigenvalues that do not satisfy these conditions, which contradicts the fact that N^k is the zero matrix.
Therefore, the correct answer is option 'c'. Zero is the only eigenvalue of a nilpotent matrix of order 4 with real entries.
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Let N be a nilpotent matrix of order 4 with real entries. Then which one of the following statements is true about eigenvalues of N?a)All eigenvalues are non zero real numbers.b)All eigenvalues are purely imaginary.c)Zero is the only eigenvalue.d)At least one eigenvalue is real and at least one eigenvalue has non zero imaginary part.Correct answer is option 'C'. Can you explain this answer?
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Let N be a nilpotent matrix of order 4 with real entries. Then which one of the following statements is true about eigenvalues of N?a)All eigenvalues are non zero real numbers.b)All eigenvalues are purely imaginary.c)Zero is the only eigenvalue.d)At least one eigenvalue is real and at least one eigenvalue has non zero imaginary part.Correct answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let N be a nilpotent matrix of order 4 with real entries. Then which one of the following statements is true about eigenvalues of N?a)All eigenvalues are non zero real numbers.b)All eigenvalues are purely imaginary.c)Zero is the only eigenvalue.d)At least one eigenvalue is real and at least one eigenvalue has non zero imaginary part.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let N be a nilpotent matrix of order 4 with real entries. Then which one of the following statements is true about eigenvalues of N?a)All eigenvalues are non zero real numbers.b)All eigenvalues are purely imaginary.c)Zero is the only eigenvalue.d)At least one eigenvalue is real and at least one eigenvalue has non zero imaginary part.Correct answer is option 'C'. Can you explain this answer?.
Let N be a nilpotent matrix of order 4 with real entries. Then which one of the following statements is true about eigenvalues of N?a)All eigenvalues are non zero real numbers.b)All eigenvalues are purely imaginary.c)Zero is the only eigenvalue.d)At least one eigenvalue is real and at least one eigenvalue has non zero imaginary part.Correct answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let N be a nilpotent matrix of order 4 with real entries. Then which one of the following statements is true about eigenvalues of N?a)All eigenvalues are non zero real numbers.b)All eigenvalues are purely imaginary.c)Zero is the only eigenvalue.d)At least one eigenvalue is real and at least one eigenvalue has non zero imaginary part.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let N be a nilpotent matrix of order 4 with real entries. Then which one of the following statements is true about eigenvalues of N?a)All eigenvalues are non zero real numbers.b)All eigenvalues are purely imaginary.c)Zero is the only eigenvalue.d)At least one eigenvalue is real and at least one eigenvalue has non zero imaginary part.Correct answer is option 'C'. Can you explain this answer?.
Solutions for Let N be a nilpotent matrix of order 4 with real entries. Then which one of the following statements is true about eigenvalues of N?a)All eigenvalues are non zero real numbers.b)All eigenvalues are purely imaginary.c)Zero is the only eigenvalue.d)At least one eigenvalue is real and at least one eigenvalue has non zero imaginary part.Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
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Let N be a nilpotent matrix of order 4 with real entries. Then which one of the following statements is true about eigenvalues of N?a)All eigenvalues are non zero real numbers.b)All eigenvalues are purely imaginary.c)Zero is the only eigenvalue.d)At least one eigenvalue is real and at least one eigenvalue has non zero imaginary part.Correct answer is option 'C'. Can you explain this answer?, a detailed solution for Let N be a nilpotent matrix of order 4 with real entries. Then which one of the following statements is true about eigenvalues of N?a)All eigenvalues are non zero real numbers.b)All eigenvalues are purely imaginary.c)Zero is the only eigenvalue.d)At least one eigenvalue is real and at least one eigenvalue has non zero imaginary part.Correct answer is option 'C'. Can you explain this answer? has been provided alongside types of Let N be a nilpotent matrix of order 4 with real entries. Then which one of the following statements is true about eigenvalues of N?a)All eigenvalues are non zero real numbers.b)All eigenvalues are purely imaginary.c)Zero is the only eigenvalue.d)At least one eigenvalue is real and at least one eigenvalue has non zero imaginary part.Correct answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
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