Question 11
(POL 3) The matrix
$$\begin{pmatrix}a_{11}&\ldots&a_{1n}\\\vdots&\cdots&\vdots\\a_{n1}&\ldots&a_{nn}\end{pmatrix}$$
satisfies the inequality $\sum_{j=1}^n|a_{j1}x_1+\cdots+a_{jn}x_n|\le M$ for each choice of numbers $x_i$ equal to $\pm1$. Show that
$$|a_{11}+a_{22}+\cdots+a_{nn}|\le M.$$
$$\begin{pmatrix}a_{11}&\ldots&a_{1n}\\\vdots&\cdots&\vdots\\a_{n1}&\ldots&a_{nn}\end{pmatrix}$$
satisfies the inequality $\sum_{j=1}^n|a_{j1}x_1+\cdots+a_{jn}x_n|\le M$ for each choice of numbers $x_i$ equal to $\pm1$. Show that
$$|a_{11}+a_{22}+\cdots+a_{nn}|\le M.$$