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Let A be 3 x 3 matrix with real entries such that det(A) = 6 and the trace of A is 0. lf det(A + I) = 0 where I denotes the 3 x 3 identity matrix, then the eigenvalues of A are
- a)-1, 2, 3
- b)-1, 2, -3
- c)1, 2, -3
- d)-1, -2, 3
Correct answer is option 'D'. Can you explain this answer?
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Let A be 3 x 3 matrix with real entries such that det(A) = 6 and the t...
Given Information:
- A is a 3x3 matrix with real entries.
- det(A) = 6
- The trace of A is 0.
- det(A + I) = 0, where I denotes the 3x3 identity matrix.
To Find:
The eigenvalues of matrix A.
Solution:
1. Trace of a Matrix:
The trace of a square matrix is the sum of its diagonal elements. Given that the trace of matrix A is 0, we have:
Trace(A) = 0
2. Eigenvalues and Determinant:
The determinant of a matrix is equal to the product of its eigenvalues. We are given that det(A) = 6. Since the matrix A is 3x3, it has three eigenvalues (possibly repeated).
3. Determinant of (A + I):
We are given that det(A + I) = 0. The determinant of a matrix is also equal to the product of its eigenvalues. So, the product of the eigenvalues of (A + I) is 0.
4. Product of Eigenvalues:
Let the eigenvalues of A be λ1, λ2, and λ3. The eigenvalues of (A + I) will be (λ1 + 1), (λ2 + 1), and (λ3 + 1). Since the product of the eigenvalues of (A + I) is 0, we have:
(λ1 + 1) * (λ2 + 1) * (λ3 + 1) = 0
5. Eigenvalues of A:
Since the product of the eigenvalues of (A + I) is 0, at least one of the eigenvalues of (A + I) must be 0. This means at least one of the eigenvalues of A must be -1.
6. Eigenvalues of A:
The determinant of A is equal to the product of its eigenvalues. We are given that det(A) = 6. Since one eigenvalue is -1, the product of the other two eigenvalues must be -6.
7. Possible Eigenvalues of A:
Considering the product of the other two eigenvalues of A is -6, and the eigenvalues are real, the possible eigenvalues are:
-1, -2, 3
-1, 2, -3
1, -2, 3
1, 2, -3
8. Eigenvalues of A:
Since the trace of A is 0, the sum of its eigenvalues is 0. Among the possible eigenvalues, only the set -1, -2, 3 satisfies this condition.
Final Answer:
Therefore, the eigenvalues of matrix A are -1, -2, and 3, which matches option 'D'.
- A is a 3x3 matrix with real entries.
- det(A) = 6
- The trace of A is 0.
- det(A + I) = 0, where I denotes the 3x3 identity matrix.
To Find:
The eigenvalues of matrix A.
Solution:
1. Trace of a Matrix:
The trace of a square matrix is the sum of its diagonal elements. Given that the trace of matrix A is 0, we have:
Trace(A) = 0
2. Eigenvalues and Determinant:
The determinant of a matrix is equal to the product of its eigenvalues. We are given that det(A) = 6. Since the matrix A is 3x3, it has three eigenvalues (possibly repeated).
3. Determinant of (A + I):
We are given that det(A + I) = 0. The determinant of a matrix is also equal to the product of its eigenvalues. So, the product of the eigenvalues of (A + I) is 0.
4. Product of Eigenvalues:
Let the eigenvalues of A be λ1, λ2, and λ3. The eigenvalues of (A + I) will be (λ1 + 1), (λ2 + 1), and (λ3 + 1). Since the product of the eigenvalues of (A + I) is 0, we have:
(λ1 + 1) * (λ2 + 1) * (λ3 + 1) = 0
5. Eigenvalues of A:
Since the product of the eigenvalues of (A + I) is 0, at least one of the eigenvalues of (A + I) must be 0. This means at least one of the eigenvalues of A must be -1.
6. Eigenvalues of A:
The determinant of A is equal to the product of its eigenvalues. We are given that det(A) = 6. Since one eigenvalue is -1, the product of the other two eigenvalues must be -6.
7. Possible Eigenvalues of A:
Considering the product of the other two eigenvalues of A is -6, and the eigenvalues are real, the possible eigenvalues are:
-1, -2, 3
-1, 2, -3
1, -2, 3
1, 2, -3
8. Eigenvalues of A:
Since the trace of A is 0, the sum of its eigenvalues is 0. Among the possible eigenvalues, only the set -1, -2, 3 satisfies this condition.
Final Answer:
Therefore, the eigenvalues of matrix A are -1, -2, and 3, which matches option 'D'.
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Let A be 3 x 3 matrix with real entries such that det(A) = 6 and the trace of A is 0. lf det(A + I) = 0 where I denotes the 3 x 3 identity matrix, then the eigenvalues of A area)-1, 2, 3b)-1, 2, -3c)1, 2, -3d)-1, -2, 3Correct answer is option 'D'. Can you explain this answer?
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Let A be 3 x 3 matrix with real entries such that det(A) = 6 and the trace of A is 0. lf det(A + I) = 0 where I denotes the 3 x 3 identity matrix, then the eigenvalues of A area)-1, 2, 3b)-1, 2, -3c)1, 2, -3d)-1, -2, 3Correct answer is option 'D'. 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 A be 3 x 3 matrix with real entries such that det(A) = 6 and the trace of A is 0. lf det(A + I) = 0 where I denotes the 3 x 3 identity matrix, then the eigenvalues of A area)-1, 2, 3b)-1, 2, -3c)1, 2, -3d)-1, -2, 3Correct answer is option 'D'. 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 A be 3 x 3 matrix with real entries such that det(A) = 6 and the trace of A is 0. lf det(A + I) = 0 where I denotes the 3 x 3 identity matrix, then the eigenvalues of A area)-1, 2, 3b)-1, 2, -3c)1, 2, -3d)-1, -2, 3Correct answer is option 'D'. Can you explain this answer?.
Let A be 3 x 3 matrix with real entries such that det(A) = 6 and the trace of A is 0. lf det(A + I) = 0 where I denotes the 3 x 3 identity matrix, then the eigenvalues of A area)-1, 2, 3b)-1, 2, -3c)1, 2, -3d)-1, -2, 3Correct answer is option 'D'. 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 A be 3 x 3 matrix with real entries such that det(A) = 6 and the trace of A is 0. lf det(A + I) = 0 where I denotes the 3 x 3 identity matrix, then the eigenvalues of A area)-1, 2, 3b)-1, 2, -3c)1, 2, -3d)-1, -2, 3Correct answer is option 'D'. 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 A be 3 x 3 matrix with real entries such that det(A) = 6 and the trace of A is 0. lf det(A + I) = 0 where I denotes the 3 x 3 identity matrix, then the eigenvalues of A area)-1, 2, 3b)-1, 2, -3c)1, 2, -3d)-1, -2, 3Correct answer is option 'D'. Can you explain this answer?.
Solutions for Let A be 3 x 3 matrix with real entries such that det(A) = 6 and the trace of A is 0. lf det(A + I) = 0 where I denotes the 3 x 3 identity matrix, then the eigenvalues of A area)-1, 2, 3b)-1, 2, -3c)1, 2, -3d)-1, -2, 3Correct answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
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