MathDB

Problem 2

Part of 2003 VJIMC

Problems(2)

sum of rows and columns of matrix, sign

Source: VJIMC 2003 1.2

7/12/2021
Let A=(aij)A=(a_{ij}) be an m×nm\times n real matrix with at least one non-zero element. For each i∈{1,…,m}i\in\{1,\ldots,m\}, let Ri=∑j=1naijR_i=\sum_{j=1}^na_{ij} be the sum of the ii-th row of the matrix AA, and for each j∈{1,…,n}j\in\{1,\ldots,n\}, let Cj=∑i=1maijC_j =\sum_{i=1}^ma_{ij} be the sum of the jj-th column of the matrix AA. Prove that there exist indices k∈{1,…,m}k\in\{1,\ldots,m\} and l∈{1,…,n}l\in\{1,\ldots,n\} such that akl>0,Rk≥0,Cl≥0,a_{kl}>0,\qquad R_k\ge0,\qquad C_l\ge0,or akl<0,Rk≤0,Cl≤0.a_{kl}<0,\qquad R_k\le0,\qquad C_l\le0.
matrixlinear algebrasuperior algebra
Disks in the Euclidean plane

Source:

5/4/2021
Let {D1,D2,...,Dn} \{D_1, D_2, ..., D_n \} be a set of disks in the Euclidean plane. Let ai,j=S(Di∩Dj) a_ {i, j} = S (D_i \cap D_j) be the area of Di∩Dj D_i \cap D_j . Prove that ∑i=1n∑j=1nai,jxixj≥0 \sum_ {i = 1} ^ n \sum_ {j = 1} ^ n a_ {i, j} x_ix_j \geq 0 for any real numbers x1,x2,...,xn x_1, x_2, ..., x_n .
linear algebra