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Collection of 4x4 matrices

Source:

September 14, 2010

linear algebramatrixprobability

Problem Statement

Let

E = \{1, 2, \dots , 16\}

and let

M

be the collection of all

4 \times 4

matrices whose entries are distinct members of

E

. If a matrix

A = (a_{ij} )_{4\times4}

is chosen randomly from

M

, compute the probability

p(k)

of

\max_i \min_j a_{ij} = k

for

k \in E

. Furthermore, determine

l \in E

such that

p(l) = \max \{p(k) | k \in E \}.

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