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I'm reading a paper where the determinant of the following matrix is solved for using some kind of recurisve method.

The matrix is given by [tex]M_{ij} = A \delta_{i,j} - B \delta_{i,j-1} - C \delta_{i,j+1}[/tex], with [tex]i,j = 1...N[/tex] and are NOT cyclic.

The author sets [tex]D_N = \texttt{det}\[M_{(N)}\][/tex] and writes the equation

[tex]D_N = A D_{N-1} - B C D_{N-2}[/tex],

assumes a solution of the form [tex]D_N = \lambda ^N[/tex] and finds two solutions,

[tex]\lambda_{\pm} = (A \pm \sqrt{A^2 - 4 B C}) / 2[/tex].

He then notes the initial conditions of [tex]D_1 = A[/tex] and [tex]D_2 = A^2 - B C[/tex] and says that the answer is therefore

[tex]D_N = \frac{\lambda_{+}^{N+1} - \lambda_{-}^{N+1}}{\lambda_{+} - \lambda_{-}}[/tex].

It's the very last step I don't understand, how did he find [tex]D_N[/tex]?

Thanks

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# Recursive solution of determinants

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