Question 38
For $n\ge2$ nonzero real numbers $a_1,a_2,\ldots,a_n$, not necessarily distinct, define the matrix $A=(a_{ij})_{1\le i,j\le n}\in M_n(\mathbb R)$ by
$$
a_{ij}=\max\{a_i,a_j\},\qquad i,j\in\{1,2,\ldots,n\}.
$$
Prove that
$$
\operatorname{rank}(A)=\left|\{a_k:k=1,2,\ldots,n\}\right|.
$$
$$
a_{ij}=\max\{a_i,a_j\},\qquad i,j\in\{1,2,\ldots,n\}.
$$
Prove that
$$
\operatorname{rank}(A)=\left|\{a_k:k=1,2,\ldots,n\}\right|.
$$