## A Relation of Nonzero Row Vectors and Column Vectors

## Problem 406

Let $A$ be an $n\times n$ matrix. Suppose that $\mathbf{y}$ is a nonzero row vector such that

\[\mathbf{y}A=\mathbf{y}.\]
(Here a row vector means a $1\times n$ matrix.)

Prove that there is a nonzero column vector $\mathbf{x}$ such that

\[A\mathbf{x}=\mathbf{x}.\]
(Here a column vector means an $n \times 1$ matrix.)