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 ______.

The correct answer is 1.

Let us analyze the properties of the identity matrix to determine its eigenvalues.
An identity matrix, denoted by
$I$
, is a square matrix where all the diagonal elements are equal to 1, and all the off-diagonal elements are equal to 0.

For any diagonal matrix (and by extension, the identity matrix), the eigenvalues are simply the entries along the main diagonal.
Since we are given a 100-dimensional identity matrix, it is a diagonal matrix of size 100 × 100 where every diagonal element is 1.

Therefore, all 100 eigenvalues of the matrix
$I$
are equal to 1.

The set
$E$
is defined as the set of its distinct real eigenvalues. Since a set contains no duplicate elements, it will only list the unique eigenvalue of the matrix:

$E=\left\{1\right\}$

Thus, the number of elements in the set
$E$
is 1.