# Determinant of sparse matrix with four nonempty subdiagonals

So I have a matrix that I want to calculate the determinant of. It is zero other than the diagonal and four subdiagonals, two on either side of the matrix. Like this:

$$\begin{bmatrix} a & & b & & & & c & & & & \\ & a & & b & & & & c& & & \\ & & \ddots& & \ddots & & & & \ddots & & \\ d&&&&&&&&&c&\\ &d&&&&&&&&&c\\ &&\ddots&&&&&&&&\\ e&&&&&&&&\ddots&&\\ &e&&&&&&&&b&\\ &&\ddots&&&\ddots&&&\ddots&&b\\ &&&e&&&d&&&a&\\ &&&&e&&&d&&&a \end{bmatrix}$$

I know in the case of one subdiagonal on either side of the matrix and the length of the diagonals are coprime, the determinant is just the product of the diagonal entries minus the product of the subdiagonal entries, but how does this generalize to two subdiagonals?

• Can we see this matrix, or is it too large to post? – Sean Roberson Aug 4 '16 at 3:59
• @SeanRoberson I included an example. – user359201 Aug 4 '16 at 4:13
• I guess you should search for "five-diagonal Toeplitz" matrices and their determinant. I am confident there should be some relevant articles. – thanasissdr Aug 4 '16 at 4:49
• @thanasissdr Thank you--what if the diagonal elements are not the same? – user359201 Aug 4 '16 at 5:51
• Your matrix hasn't four sub-diagonals. It has one main diagonal, two sub-diagonals and two super-diagonals. – user1551 Aug 4 '16 at 8:01