Lehmer matrix

In mathematics, particularly matrix theory, the "n×n" Lehmer matrix (named after Derrick Henry Lehmer) is the constant symmetric matrix defined by
Properties.
As can be seen in the examples section, if "A" is an "n×n" Lehmer matrix and "B" is an "m×m" Lehmer matrix, then "A" is a submatrix of "B" whenever "m">"n". The values of elements diminish toward zero away from the diagonal, where all elements have value 1.
The inverse of a Lehmer matrix is a tridiagonal matrix, where the superdiagonal and subdiagonal have strictly negative entries. Consider again the "n×n" "A" and "m×m" "B" Lehmer matrices, where "m">"n". A rather peculiar property of their inverses is that "A−1" is "nearly" a submatrix of "B−1", except for the "A−1n,n" element, which is not equal to "B−1n,n".
A Lehmer matrix of order "n" has trace "n".
Examples.
The 2×2, 3×3 and 4×4 Lehmer matrices and their inverses are shown below.