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Let $A$ be the $n\times n$ matrix given by

$$A_{ij}=\frac{1}{i + j - 1}$$

Show that $A$ is invertible and that the inverse has integer entries.

I was able to show that $A$ is invertible. How do I show that $A^{-1}$ has integer entries?

This matrix is called the Hilbert matrix. The problem appears as exercise 12 in section 1.6 of Hoffman and Kunze's Linear Algebra (2nd edition).

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user52991
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    In fact, you have a closed form for the entries of $A^{-1}$. This is a Hilbert matrix – Ewan Delanoy Jun 26 '13 at 14:58
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    May I ask how you proved that $A$ is invertible? I do remember trying to find an elegant proof of that about a year ago and didn't quite manage to find it. (Just to be sure: I don't consider it elegant in this case to find the determinant) – HSN Jun 26 '13 at 15:10
  • I forgot to add to the bounty description that a brief overview of why the matrix is invertible would be nice too. – Potato Jul 29 '13 at 16:50
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    Related question on MO: http://mathoverflow.net/questions/47561/deriving-inverse-of-hilbert-matrix – Jason Conti Jul 29 '13 at 16:53
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    Related, perhaps dupe of http://math.stackexchange.com/q/46862/8271 – leo Jul 29 '13 at 18:42
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    @HSN The vectors $x^i$ in $L_2[0,1]$ are linearly independent, so their Gram matrix is invertible. –  Jul 31 '13 at 16:47

1 Answers1

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Be wise, generalize (c)

I think the nicest way to answer this question is the direct computation of the inverse - however, for a more general matrix including the Hilbert matrix as a special case. The corresponding formulas have very transparent structure and nontrivial further generalizations.

The matrix $A$ is a particular case of the so-called Cauchy matrix with elements $$A_{ij}=\frac{1}{x_i-y_j},\qquad i,j=1,\ldots, N.$$ Namely, in the Hilbert case we can take $$x_i=i-\frac{1}{2},\qquad y_i=-i+\frac12.$$ The determinant of $A$ is given in the general case by $$\mathrm{det}\,A=\frac{\prod_{1\leq i<j\leq N}(x_i-x_j)(y_j-y_i)}{\prod_{1\leq i,j\leq N}(x_i-y_j)}.\tag{1}$$ Up to an easily computable constant prefactor, the structure of (1) follows from the observation that $\mathrm{det}\,A$ vanishes whenever there is a pair of coinciding $x$'s or $y$'s. (In the latter case $A$ contains a pair of coinciding raws/columns). For our $x$'s and $y$'s the determinant is clearly non-zero, hence $A$ is invertible.

One can also easily find the inverse $A^{-1}$, since the matrix obtained from a Cauchy matrix by deleting one row and one column is also of Cauchy type, with one $x$ and one $y$ less. Taking the ratio of the corresponding two determinants and using (1), most of the factors cancel out and one obtains \begin{align} A_{mn}^{-1}=\frac{1}{y_m-x_n}\frac{\prod_{1\leq i\leq N}(x_n-y_i)\cdot\prod_{1\leq i\leq N}(y_m-x_i)}{\prod_{i\neq n}(x_n-x_i)\cdot\prod_{i\neq m}(y_m-y_i)}.\tag{2} \end{align}

For our particular $x$'s and $y$'s, the formula (2) reduces to \begin{align} A_{mn}^{-1}&=\frac{(-1)^{m+n}}{m+n-1}\frac{\frac{(n+N-1)!}{(n-1)!}\cdot \frac{(m+N-1)!}{(m-1)!}}{(n-1)!(N-n)!\cdot(m-1)!(N-m)!}=\\ &=(-1)^{m+n}(m+n-1){n+N-1 \choose N-m}{m+N-1 \choose N-n}{m+n-2\choose m-1}^2. \end{align} The last expression is clearly integer. $\blacksquare$

darij grinberg
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    Thanks for this. The only problem I have with this answer is that it is asked in Chapter 1 of Hoffman and Kunze long before the introduction of the concept of determinant. All we know really is that an invertible matrix can be row-reduced to the identity. – Gregory Grant Jun 21 '16 at 19:11
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    @GregoryGrant And did you find any solution just using the tools provided for the first exercise? I'm curious. – Bijesh K.S May 07 '17 at 18:55
  • A surprising fact is that it can be proven that the sum of the entries of the inverse is $n^2$, which yields a very unusual identity. – Jose Pérez Cano May 15 '22 at 09:47