Row and Column Sums: Square Matrices and Nilpotence
Company: Voleon
Role: Quantitative Researcher
Category: Statistics & Math
Difficulty: medium
Interview Round: Technical Screen
A finite matrix has every row sum equal to 1 and every column sum equal to 1. Must the matrix be square? If so, can it be nilpotent?
### Constraints & Assumptions
The matrix has a positive number of rows and columns and real entries. Entries need not be nonnegative. Nilpotent means that some positive integer power of a square matrix equals the zero matrix.
### Clarifying Questions
What is the sum of all entries when counted by rows and by columns? What happens when the matrix multiplies the all-ones vector?
### What a Strong Answer Covers
A short proof of the shape constraint and a separate argument ruling out nilpotence, without assuming unstated positivity.
### Follow-up Questions
Would the conclusion change if row sums were r and column sums were c instead? Why is the all-ones vector argument stronger than inspecting a few powers?
Overview: Prove that a real matrix with all row and column sums equal to one is square, then use the all-ones eigenvector to rule out nilpotence.
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