MathDB

For

n\geq 0

define the matrices

A_n

and

B_n

as follows: A_0 \equal{} B_0 \equal{} (1), and for every

n>0

let A_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\ A_{n \minus{} 1} & B_{n \minus{} 1} \\ \end{array} \right) \ \textrm{and} \ B_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\ A_{n \minus{} 1} & 0 \\ \end{array} \right). Denote by

S(M)

the sum of all the elements of a matrix

M

. Prove that S(A_n^{k \minus{} 1}) \equal{} S(A_k^{n \minus{} 1}), for all

n,k\geq 2

Problem Statement