Let I be a 100 dimensional identity matrix and E be the set of its dis...
I
_{100}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.