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Let I be a 100 dimensional identity matrix and E be the set of its distinct (no value appears more than once in E) real eigenvalues. The number of elements in E is ______.
Correct answer is '1'. Can you explain this answer?
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Let I be a 100 dimensional identity matrix and E be the set of its dis...
I100
Eigen values of I →
Set of distributed eigen value E = {1} Number of elements in E = 1
Eigen values of I →
Set of distributed eigen value E = {1} Number of elements in E = 1
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Let I be a 100 dimensional identity matrix and E be the set of its dis...
Explanation:
To find the distinct real eigenvalues of the 100-dimensional identity matrix I, we need to understand the properties of the identity matrix.
Identity Matrix:
An identity matrix is a square matrix in which all the elements of the principal diagonal (top-left to bottom-right) are ones, and all other elements are zeros. It is denoted by I.
Properties of Identity Matrix:
1. The identity matrix is always a square matrix.
2. The identity matrix is a diagonal matrix.
3. The identity matrix is always invertible.
4. The identity matrix has eigenvalues equal to 1 and eigenvectors equal to the standard basis vectors.
Eigenvalues and Eigenvectors:
Eigenvalues are scalars that represent the scale factor of the eigenvectors when a linear transformation is applied to them. Eigenvectors are non-zero vectors that remain in the same direction but may be scaled by a scalar value, which is the eigenvalue.
Distinct Eigenvalues:
Distinct eigenvalues are unique eigenvalues that do not repeat in the set of eigenvalues.
Solution:
Since the identity matrix I is a diagonal matrix, all the diagonal elements are equal to 1. Thus, the characteristic equation for finding the eigenvalues can be written as:
det(I - λI) = 0
where I is the identity matrix and λ is the eigenvalue.
Substituting the values, we get:
det(I - λI) = det((1 - λ)I) = (1 - λ)^100 = 0
For the determinant to be zero, (1 - λ) must be zero, i.e., λ = 1.
Therefore, the only eigenvalue of the 100-dimensional identity matrix I is 1. As there is only one distinct eigenvalue, the number of elements in the set E is 1.
Hence, the correct answer is 1.
To find the distinct real eigenvalues of the 100-dimensional identity matrix I, we need to understand the properties of the identity matrix.
Identity Matrix:
An identity matrix is a square matrix in which all the elements of the principal diagonal (top-left to bottom-right) are ones, and all other elements are zeros. It is denoted by I.
Properties of Identity Matrix:
1. The identity matrix is always a square matrix.
2. The identity matrix is a diagonal matrix.
3. The identity matrix is always invertible.
4. The identity matrix has eigenvalues equal to 1 and eigenvectors equal to the standard basis vectors.
Eigenvalues and Eigenvectors:
Eigenvalues are scalars that represent the scale factor of the eigenvectors when a linear transformation is applied to them. Eigenvectors are non-zero vectors that remain in the same direction but may be scaled by a scalar value, which is the eigenvalue.
Distinct Eigenvalues:
Distinct eigenvalues are unique eigenvalues that do not repeat in the set of eigenvalues.
Solution:
Since the identity matrix I is a diagonal matrix, all the diagonal elements are equal to 1. Thus, the characteristic equation for finding the eigenvalues can be written as:
det(I - λI) = 0
where I is the identity matrix and λ is the eigenvalue.
Substituting the values, we get:
det(I - λI) = det((1 - λ)I) = (1 - λ)^100 = 0
For the determinant to be zero, (1 - λ) must be zero, i.e., λ = 1.
Therefore, the only eigenvalue of the 100-dimensional identity matrix I is 1. As there is only one distinct eigenvalue, the number of elements in the set E is 1.
Hence, the correct answer is 1.
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Let I be a 100 dimensional identity matrix and E be the set of its distinct (no value appears more than once in E) real eigenvalues. The number of elements in E is ______.Correct answer is '1'. Can you explain this answer?
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Let I be a 100 dimensional identity matrix and E be the set of its distinct (no value appears more than once in E) real eigenvalues. The number of elements in E is ______.Correct answer is '1'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Let I be a 100 dimensional identity matrix and E be the set of its distinct (no value appears more than once in E) real eigenvalues. The number of elements in E is ______.Correct answer is '1'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let I be a 100 dimensional identity matrix and E be the set of its distinct (no value appears more than once in E) real eigenvalues. The number of elements in E is ______.Correct answer is '1'. Can you explain this answer?.
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