每週問題 March 25, 2013

Let $A=[a_{ij}]$ be an $n\times n$ matrix with $a_{ij}=x_i+y_j$. Show that $\mathrm{rank}A\le 2$.

$X=\begin{bmatrix} x_1&1\\ x_2&1\\ \vdots&\vdots\\ x_n&1 \end{bmatrix},~~Y=\begin{bmatrix} 1&1&\cdots&1\\ y_1&y_2&\cdots&y_n \end{bmatrix}$

PowSol-March-25-13

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