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Another approach: the determinant of a matrix is the product of its eigenvalues. The matrix J is rank 1 and so has only one non-zero eigenvalue, which is k (with eigenvector proportional the vector of all ones). Thus cJ has one non-zero eigenvalue, which is ck. Adding I to a matrix adds 1 to all eigenvalues, so I+cJ has k-1 eigenvalues being 1 and the remaining one being 1+ck. The matrix a(I+cJ) has k-1 eigenvalues being a and the remaining one being a(1+ck). Multiplying these k eigenvalues and simplifying gives the desires result. Best wishes, Tobias Rydén Kim T. Parker wrote: > Stefano, > > If you divide every element of the matrix by beta, you end up with > alpha/beta on the diagonals and 1 everywhere else. If we write > c=alpha/beta, the determinant of the matrix is easier to find - you > subtract the second row from the first row which gives you only two > non-zero elements in the first row and thus only two terms in the > expansion of the determinant. You can then spot some recurrence > relationships between the k by k matrix and the (k-1) by (k-1) > cofactors. Following these through gives the determinant as being > (c-1)^(k-1)*(c+k-1). However, we divided every element in the k rows > of the original matrix by beta, so we have to multiply our simplified > determinant by beta^k. Applying this factor, and substituting for c > in terms of alpha and beta, gives the following answer: > > (alpha - beta)^(k-1)*(alpha + beta*(k - 1)) > > I've checked this up to beta=4 so it looks all right. Other > Allstatters might spot a flaw, or have a more elegant way of deriving > the result. > > Best wishes, > > Kim Parker > > Stefano Sofia wrote: > >> Dear Allstat users, >> probably some of you will be able to help me. >> I have a square matrix of dimension k by k with alpha on the diagonal >> and beta everywhee else. This symmetric matrix is called symmetric >> compound matrix and has the form a( I + cJ), where I is the k by k >> identity matrix J is the k by k matrix of all ones >> a = alpha - beta >> c = beta/a >> >> I need to evaluate the determinant of this matrix. Is there any >> algebric formula for that? >> >> thank you for your help >> Stefano >> >> > -- -- Tobias Rydén E-mail: [log in to unmask] Centre for Mathematical Sciences Tel: int+46-46 222 4778 Lund University Fax: int+46-46 222 4623 -- Box 118, S-221 00 Lund, Sweden WWW: www.maths.lth.se/matstat