I was asked to find to show that a matrix $A= (a_{i,j}) \in M_n(\mathbb R)$ such that
$$\forall i, \sum_{j=1}^n |a_{i,j}|\le 1 \quad (1)$$
satisfies $$|\det(A)|\le 1, \quad (2)$$ and then to find the matrices $A$ that satisfy $(1)$ and the equality in $(2)$.
The first part was not difficult because $(1)$ means that for all $X=(x_i)\in \mathbb C^n$, we have
$$\| AX\|_\infty \le \| X\|_\infty$$
where $\| X\|_\infty = \max_i |x_i|$, hence the spectral radius of $A$ is $\le 1$, and $|\det(A)|\le 1$.
For the second part of the question, if $A$ satisfies $(1)$ and
$$\exists i_0, \sum_{j=1}^n |a_{i_0,j}|=\alpha <1,$$
then the matrix obtained by multiplying the $i_0$-th row of $A$ by $\alpha^{-1}$, has determinant equal to $\alpha^{-1} \det(A) \le 1$, then $ \det(A) \le \alpha <1$. Hence if $A$ satisfies $(1)$ and $(2)$, then
$$\forall i, \sum_{j=1}^n |a_{i,j}| =1. \quad $$
I found that the permutation matrices satisfy the equality in $(2)$, and by taking a permutation matrix and replacing some of the coefficients equal to $1$ by $-1$, we also obtain a matrix that satisfies the equality in $(2)$, but I'm stuck here.
This should get you started. Note that for a set $K$ of finite $n$-dimensional volume in $\mathbb R^n$, the volume of $AK$ is $|\det(A)|$ times the volume of $K$. In particular let $K = \{x \in \mathbb R^n: \|x\|_\infty \le 1\}$. Since $A K \le K$ and they have the same volume, $AK = K$. The extreme points of $K$ are $\{-1,1\}^n$, and $A$ must permute these.