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IMC 1994 D1 P1

Source:

March 6, 2017
IMClinear algebramatrix

Problem Statement

a) Let AA be a n×nn\times n, n≥2n\geq 2, symmetric, invertible matrix with real positive elements. Show that zn≤n2−2nz_n\leq n^2-2n, where znz_n is the number of zero elements in A−1A^{-1}.
b) How many zero elements are there in the inverse of the n×nn\times n matrix A=(1111…11222…21211…11212…2⋮⋮⋮⋮⋱⋮1212…⋱)A=\begin{pmatrix} 1&1&1&1&\ldots&1\\ 1&2&2&2&\ldots&2\\ 1&2&1&1&\ldots&1\\ 1&2&1&2&\ldots&2\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 1&2&1&2&\ldots&\ddots \end{pmatrix}
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