# Problems with the inverse of a banded matrix: not invertible?

I am creating with a software a banded matrix, which is also symmetric. In fact, its definition comes from an array, Array[q], whose length is n, where I store my values. Then to define my matrix I do the following:

Matrix[i][j] = Array[abs[i-j]],

where abs indicates the absolute value. The result is, in the case of n=8 for example,

{0.590818, 0.574632, 0.528687, 0.460129, 0.378821, 0.295026,
0.21735, 0.151471},
{0.574632, 0.590818, 0.574632, 0.528687, 0.460129, 0.378821,
0.295026, 0.21735},
{0.528687, 0.574632, 0.590818, 0.574632, 0.528687, 0.460129,
0.378821, 0.295026},
{0.460129, 0.528687, 0.574632, 0.590818, 0.574632, 0.528687,
0.460129, 0.378821},
{0.378821, 0.460129, 0.528687, 0.574632, 0.590818, 0.574632,
0.528687, 0.460129},
{0.295026, 0.378821, 0.460129, 0.528687, 0.574632, 0.590818,
0.574632, 0.528687},
{0.21735, 0.295026, 0.378821, 0.460129, 0.528687, 0.574632,
0.590818, 0.574632},
{0.151471, 0.21735, 0.295026, 0.378821, 0.460129, 0.528687,
0.574632, 0.590818}
}


Now the point is that I need to calculate the inverse of this matrix. As long as n=8, I manage to make the software do that, but as soon as I grow with the dimension, say n=64 for example, I get the error that the matrix is badly conditioned, because the determinant is (almost) 0. In fact, if I calculate the determinant in the case n=64, I get that it is equal to

1.483771072856803*10^-498


My question is: why should such a banded matrix be singular? I cannot see an evident reason.

• It seems to me that Matrix[i][j]=Array[abs[i-j]] defines a symmetric Toeplitz matrix instead of a general symmetric banded matrix. – Algebraic Pavel Apr 16 '15 at 10:01
• A small comment: I don't know what software do you use but if it determines bad conditioning by looking at the value of the determinant, I would suggest replacing it by another software since determinant has no connection to the condition number whatsoever. For example, a diagonal $n\times n$ matrix $D:=\mathrm{diag}(0.1,\ldots,0.1)$ has determinant $10^{-n}$ (arbitrarily small for large $n$) but its spectral condition number is 1. – Algebraic Pavel Apr 16 '15 at 10:03