MathDB

1

Part of 1994 IMC

Problems(2)

IMC 1994 D1 P1

Source:

3/6/2017
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}
IMClinear algebramatrix
IMC 1994 D2 P1

Source:

3/6/2017
Let f∈C1[a,b]f\in C^1[a,b], f(a)=0f(a)=0 and suppose that λ∈R\lambda\in\mathbb R, λ>0\lambda >0, is such that ∣f′(x)∣≤λ∣f(x)∣|f'(x)|\leq \lambda |f(x)| for all x∈[a,b]x\in [a,b]. Is it true that f(x)=0f(x)=0 for all x∈[a,b]x\in [a,b]?
IMCreal analysis