# Question about an inequality described by matrices

$$A = (a_ {ij}) _ {1 le i, j le n}$$ is a matrix that$$sum_ limits {i = 1} ^ {n} a_ {ij} = 1$$ for every j and $$sum_ limits {j = 1} ^ n a_ {ij} = 1$$ for every i and $$a_ {ij} ge 0$$.And
$$begin {equation} begin {pmatrix} y_1 \ vdots \ y_n \ {pmatrix} = mathbf {A} begin {pmatrix} x_1 \ vdots \ x_n {pmatrix} end {equation}$$
$$y_i$$ and $$x_i$$ are not all negative. Prove: $$y_1 cdots y_n ge x_1 cdots x_n$$

It can somehow be important to convexize the function.