Tagged: nonzero vector

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.)

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