## A Singular Matrix and Matrix Equations $A\mathbf{x}=\mathbf{e}_i$ With Unit Vectors

## Problem 561

Let $A$ be a singular $n\times n$ matrix.

Let

\[\mathbf{e}_1=\begin{bmatrix}

1 \\

0 \\

\vdots \\

0

\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}

0 \\

1 \\

\vdots \\

0

\end{bmatrix}, \dots, \mathbf{e}_n=\begin{bmatrix}

0 \\

0 \\

\vdots \\

1

\end{bmatrix}\]
be unit vectors in $\R^n$.

Prove that at least one of the following matrix equations

\[A\mathbf{x}=\mathbf{e}_i\]
for $i=1,2,\dots, n$, must have no solution $\mathbf{x}\in \R^n$.