MathDB

2

Part of 1997 IMC

Problems(2)

sum convergence

Source: IMC 1997 day 1 problem 2

10/1/2005
Let ana_n be a sequence of reals. Suppose ∑an\sum a_n converges. Do these sums converge aswell? (a) a1+a2+(a4+a3)+(a8+...+a5)+(a16+...+a9)+...a_1+a_2+(a_4+a_3)+(a_8+...+a_5)+(a_{16}+...+a_9)+... (b) a1+a2+(a3)+(a4)+(a5+a7)+(a6+a8)+(a9+a11+a13+a15)+(a10+a12+a14+a16)+(a17+a19+...{a_1+a_2+(a_3)+(a_4)+(a_5+a_7)+(a_6+a_8)+(a_9+a_{11}+a_{13}+a_{15})+(a_{10}+a_{12}+a_{14}+a_{16})+(a_{17}+a_{19}+...}
real analysisreal analysis unsolved
Det of block matrix (trivial)

Source: IMC 1997 day 2 problem 2

10/19/2005
Let M∈GL2n(K)M \in GL_{2n}(K), represented in block form as M=[ABCD],M−1=[EFGH] M = \left[ \begin{array}{cc} A & B \\ C & D \end{array} \right] , M^{-1} = \left[ \begin{array}{cc} E & F \\ G & H \end{array} \right] Show that det⁡M.det⁡H=det⁡A\det M.\det H=\det A.
linear algebramatrixlinear algebra unsolved