Linear Algebra Done Right · Problem 3C.13

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · matrix-multiplication

Problem

Suppose

A

is an

n

-by-

n

matrix and

1 \leq j, k \leq n

. Show that the entry in row

j

, column

k

, of

A^3

(which is defined to mean

AAA

) is

\sum_{p=1}^{n} \sum_{r=1}^{n} A_{j,p} A_{p,r} A_{r,k}.