Wednesday, 24 April 2019

matrices - Determinant of NtimesN matrix



I have the following matrix which I need to find the determinant of. I am not too sure of how to proceed. Here is my working so far.




det(J(Ei)λI)=(A1λϕ1i0ci1AiλciN0ϕNiANλ)


The matrix has a diagonal given by Ajλ. From the central red element there are vertically and horizontally non-zero elements. All other elements are zero exactly.



I am really not sure how to find the determinant from here and any help or pointers would be greatly appreciated!



Edit



For instance if N where to equal 4 we might have the following case if i=3,
det(J(Ei)λI)=(D10V100D2V20H1H2D3H400V4D4)


Answer



Edit. Let B the submatrix obtained by deleting the i-th row and i-th column of the given matrix. Then the required determinant is the product of determinant of the Schur complement of B and detB, i.e.
(Aiλ+kickϕkiAkλ)ki(Akλ).


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