TMMATE 2009, Problema 3

[Laurian Filip]

Pentru \( n \in \mathbb{N}^* \) se considera \( A=(a_{ij}) \) o matrice cu proprietatea ca \( a_{ij}\in \mathbb{Z} \) si \( |a_{ij}|=\max \lbrace i,j \rbrace, \forall i,j \in \overline{1,n} \). Aratati ca:
a) Daca n este impar, atunci \( \det A \) este impar;
b) Matricea A este inversabila pentru orice \( n\in \mathbb{N}^* \).

Andrei Eckstein

[Theodor Munteanu]

\(
\left| {\begin{array}
1 & 2 & 3 & {...} & n \\
2 & 2 & 3 & {...} & n \\
3 & 3 & 3 & {...} & n \\
{...} & {...} & {...} & {...} & {...} \\
n & n & n & {...} & n \\
\end{array}} \right| = n\left| {\begin{array}
1 & 2 & 3 & {...} & n \\
2 & 2 & 3 & {...} & n \\
3 & 3 & 3 & {...} & n \\
{...} & {...} & {...} & {...} & {...} \\
1 & 1 & 1 & {...} & 1 \\
\end{array}} \right| = nZ
\);
Pentru n impar Z\( \equiv \limits_{\bmod 2} \left| {\begin{array}
1 & 0 & 1 & 0 & {...} & 0 & 1 \\
0 & 0 & 1 & 0 & {...} & 0 & 1 \\
1 & 1 & 1 & 0 & {...} & 0 & 1 \\
0 & 0 & 0 & 0 & {...} & 0 & 1 \\
{...} & {...} & {...} & {...} & {...} & {...} & {...} \\
0 & 0 & 0 & 0 & {...} & 0 & 1 \\
1 & 1 & 1 & 1 & {...} & 1 & 1 \\
\end{array}} \right| = ( - 1)\left| {\begin{array}
1 & 0 & 1 & 0 & {...} & 1 & 0 \\
0 & 0 & 1 & 0 & {...} & 1 & 0 \\
1 & 1 & 1 & 0 & {...} & 1 & 0 \\
0 & 0 & 0 & 0 & {...} & 1 & 0 \\
{...} & {...} & {...} & {...} & {...} & {...} & {...} \\
1 & 1 & 1 & 1 & {...} & 1 & 0 \\
1 & 1 & 1 & 1 & {...} & 1 & 1 \\
\end{array}} \right| \);
Daca notam prima matrice de ordin n cu \( {\rm A}_{\rm n} \), atunci \(
\left| {{\rm A}_{\rm n} } \right| = - \left| {A_{n - 2} } \right| \);
\( \det A_1 = 1 \Rightarrow \det A_n \) e impar;
Daca n e par, atunci \( {\rm Z} \equiv {\rm det B}_{\rm n} \) unde
\(
{\rm det B}_{\rm n} = \left| {\begin{array}
1 & 0 & 1 & 0 & {...} & 1 & 0 \\
0 & 0 & 1 & 0 & {...} & 1 & 0 \\
1 & 1 & 1 & 0 & {...} & 1 & 0 \\
0 & 0 & 0 & 0 & {...} & 1 & 0 \\
{...} & {...} & {...} & {...} & {...} & {...} & {...} \\
1 & 1 & 1 & 1 & {...} & 1 & 0 \\
1 & 1 & 1 & 1 & {...} & 1 & 1 \\
\end{array}} \right| = \det A_{n - 1}
\) adica impar deci diferit de 0.