Lehmer matrix

In mathematics, particularly matrix theory, the n×n Lehmer matrix (named after Derrick Henry Lehmer) is the constant symmetric matrix defined by
 * $$A_{ij} =

\begin{cases} i/j, & j\ge i \\ j/i, & jn. 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.

\begin{array}{lllll} A_2=\begin{pmatrix} 1  & 1/2  \\  1/2 &   1  \end{pmatrix}; & A_2^{-1}=\begin{pmatrix} 4/3 & -2/3 \\ -2/3 & {\color{Brown}{\mathbf{4/3}}} \end{pmatrix};

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A_3=\begin{pmatrix} 1  & 1/2 & 1/3 \\  1/2 &   1 & 2/3 \\  1/3 & 2/3 &   1 \end{pmatrix}; & A_3^{-1}=\begin{pmatrix} 4/3 & -2/3 &      \\ -2/3 & 32/15 & -6/5 \\      & -6/5  & {\color{Brown}{\mathbf{9/5}}} \end{pmatrix};

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A_4=\begin{pmatrix} 1  & 1/2 & 1/3 & 1/4 \\  1/2 &   1 & 2/3 & 1/2 \\  1/3 & 2/3 &   1 & 3/4 \\  1/4 & 1/2 & 3/4 & 1 \end{pmatrix}; & A_4^{-1}=\begin{pmatrix} 4/3 & -2/3 &        &       \\ -2/3 & 32/15 &  -6/5  &       \\      & -6/5  & 108/35 & -12/7 \\      &       & -12/7  & {\color{Brown}{\mathbf{16/7}}} \end{pmatrix}. \\ \end{array} $$