Row and Column Sums: Square Matrices and Nilpotence

Consider a finite matrix where each row sum is 1 and each column sum is 1. Is the matrix necessarily square? If yes, could it be nilpotent?

Constraints & Assumptions

Assume the matrix has at least one row and at least one column, and its entries are real numbers. The entries are not required to be nonnegative. Nilpotent means some positive integer power of the (square) matrix yields the zero matrix.

Clarifying Questions

What is the total sum of all entries computed first by row sums and then by column sums? What is the result of multiplying the matrix by the all-ones vector?

What a Strong Answer Covers

A concise proof that the matrix must be square, plus a separate argument that nilpotence is impossible, without assuming any unstated positivity conditions.

Follow-up Questions

Would the conclusion be different if the row sums were all r and the column sums all c, instead of 1? Why is the all-ones vector argument stronger than checking just a few powers?

Overview: Prove that a real matrix with every row and column sum equal to 1 must be square, and then use the all-ones eigenvector to rule out nilpotence.