MathDB

Let

k

and

n

be positive integers. A sequence

\left( A_1, \dots , A_k \right)

n\times n

real matrices is preferred by Ivan the Confessor if

A_i^2\neq 0

for

1\le i\le k

, but

A_iA_j=0

for

1\le i

j\le k

with

i\neq j

. Show that

k\le n

in all preferred sequences, and give an example of a preferred sequence with

k=n

for each

n

.
(Proposed by Fedor Petrov, St. Petersburg State University)

Problem Statement