The matrix
![A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}](/media/m/b/7/9/b798db655fa4ebc7f113ee97e487f510.png)
satisfies the inequality
![\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M](/media/m/d/f/4/df4d6a039fb46db78a8b959b0c36eefe.png)
for each choice of numbers
![x_i](/media/m/c/2/8/c28855cc6fcd9627da560688b31cda10.png)
equal to
![\pm 1](/media/m/f/4/4/f442dcf468f29c14c07a53c982f321a6.png)
. Show that
%V0
The matrix
$$A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}$$
satisfies the inequality $\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M$ for each choice of numbers $x_i$ equal to $\pm 1$. Show that
$$|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.$$