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Is there any fast way to compute the determinant of this matrix:

$ \begin{vmatrix} a & b & 0 &0 &0 \\ b & a & b &0 &0 \\ 0 & b & a &b &0 \\ 0 & 0 & b &a &b \\ 0 & 0 & 0 &b &a \end{vmatrix} $

And can you say anything about $ \frac{det(M_{k+1})}{det(M_{k})} $

where $M_{k}$ is a toeplitz matrix?

2 Answers 2

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If $T_n$ is the determinant for $n$x$n$ matrix, then we have by expanding using the first row (and first column in the second matrix) that

$T_n = a T_{n-1} - b^2 T_{n-2}$

with $T_1 = a$ and $T_2 = a^2 - b^2$.

You should now be able to compute a closed form easily, using the characteristic equation.

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    @neticin: See: http://mathworld.wolfram.com/DeterminantExpansionbyMinors.html. We pick the first row to get two minors, in the second minor, we pick the first column.2012-02-09
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See the paper Analytical Inversion of Symmetric Tridiagonal Matrices, and in particular the formulas (6) and (7). To apply this it is only necessary to remove a factor of $b$ from each row of your matrix, with the accompanying factor of $b^n$ from the determinant.

Of course if $b = 0$...