# matrix properties if all the solutions are the same

Assume I have linear equation system in $n$ variables and the solution is $x_1=x_2=\cdots = x_n$.

What properties must the matrix of the system have, or how do I prove that the system has such a solution?

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I assume that you have an $n\times n$ linear system $Ax=0$, whose solutions are precisely $x_1=x_2=\cdots=x_n$. This means that the null space is the linear span of the vector $e=(1,1,\cdots,1)$. So the row space is the orthogonal complement of $e$. That means the sum of each row of $A$ must be 0, and that $n-1$ of these rows must be linearly independent.