Let $n \ge 2$ and $ a_1, a_2, \ldots , a_n $, nonzero real numbers not necessarily distinct. We define matrix $A = (a_{ij})_{1 \le i,j \le n} \in M_n( \mathbb{R} )$ , $a_{i,j} = max \{ a_i, a_j \}$, $\forall i,j \in \{ 1,2 , \ldots , n \} $. Show that $\mathbf{rank}(A) $= $\mathbf{card} $ $\{ a_k | k = 1,2, \ldots n \} $
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Tags: linear algebra
Acridian9
29.04.2021 08:40
DanDumitrescu wrote: Let $n \ge 2$ and $ a_1, a_2, \ldots , a_n $, real numbers not necessarily distinct. We define matrix $A = (a_{ij})_{1 \le i,j \le n} \in M_n( \mathbb{R} )$ , $a_{i,j} = max \{ a_i, a_j \}$, $\forall i,j \in \{ 1,2 , \ldots , n \} $. Show that $\mathbf{rank}(A) $= $\mathbf{card} $ $\{ a_k | k = 1,2, \ldots n \} $ There's something wrong here. What about if $\max \{ a_1,\ldots, a_n \}=0$? For instance, $A$ can never have rank $=n$, even if the $a_i$ are distinct.
Filipjack
29.04.2021 09:28
Indeed, in the original statement it is mentioned that $a_1,a_2, \ldots, a_n$ are nonzero real numbers.
ZETA_in_olympiad
04.03.2022 09:05
As far as I know, Grade 11 MO in Romania is not college.