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1. ### Cauchy expansion of determinant of a bordered matrix

Hi! It just dawned on me that any such matrices (I suppose there are only 4 places A could go ^^, ) are related by simple permutations. Since any permutation matrix has determinant + or - 1 then what you say must be true. Thank you for the enlightenment! =)
2. ### Cauchy expansion of determinant of a bordered matrix

The Cauchy expansion says that \text{det} \begin{bmatrix} A & x \\[0.3em] y^T & a \end{bmatrix} = a \text{det}(A) - y^T \text{adj}(A) x , where A is an n-1 by n-1 matrix, y and x are vectors with n-1 elements, and a is a scalar. There is a proof in Matrix Analysis by Horn and...
3. ### Addition to a random matrix element

Hi all! I have no application in mind for the following question but it find it curious to think about: Say that we have a square matrix where the sum of the elements in each row and each column is zero. Clearly such a matrix is singular. Suppose that no row or column of the matrix is the...
4. ### Properties of a special block matrix

Thank you! I think I shall have to return to the drawing board for a closer investigation :)
5. ### Properties of a special block matrix

That's a fair point. I was playing around with a different matrix - Hermitian and also symmetric about the anti-diagonal. Turns out that the eigenvectors are closely related to the eigenvectors of the matrix R above so I was curious about their structure. It seems reasonable that the upper and...
6. ### Properties of a special block matrix

Hi folks! I've encountered the matrix below and I'm curious about its properties; R= \begin{pmatrix} 0 & N-S\\ N+S & 0 \end{pmatrix} where R, N and S are real matrices, R is 2n by 2n, N is n by n symmetric and S is n by n skew-symmetric. Clearly R is symmetric so the...
7. ### Zero as an element of an eigenvector

Thank you for your reply! Is there a way to determine from the matrix whether a zero will appear without calculating the eigenvector explicitly?
8. ### Zero as an element of an eigenvector

Quick question on eigenvectors; Are there any general properties of a matrix that guarantee that a zero will or will not appear as an element in an eigenvector? Thank you!