Dissertativa 6 - 2ª Fase - Dia 1 - ITA 2026

$$\begin{gathered} \mathrm{A}=\left[\begin{array}{cccccc}0& 0& 0& 0& \cdots & 0\\ {\mathrm{\alpha }}_1^0& 0& 0& 0& \cdots & 0\\ {\mathrm{\alpha }}_1^1& {\mathrm{\alpha }}_2^0& 0& 0& \cdots & 0\\ {\mathrm{\alpha }}_1^2& {\mathrm{\alpha }}_2^1& {\mathrm{\alpha }}_3^0& 0& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ {\mathrm{\alpha }}_1^{\mathrm{n}-2}& {\mathrm{\alpha }}_2^{\mathrm{n}-3}& {\mathrm{\alpha }}_3^{\mathrm{n}-4}& {\mathrm{\alpha }}_4^{\mathrm{n}-5}& \cdots & 0\end{array}\right], \mathrm{B}=\left[\begin{array}{cccccc}1& 1& 1& 1& \cdots & 1\\ 0& 1& 1& 1& \cdots & 1\\ 0& 0& 1& 1& \cdots & 1\\ 0& 0& 0& 1& \cdots & 1\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& 0& \cdots & 1\end{array}\right] \\ \mathrm{C}=\mathrm{AB}=\left[\begin{array}{cccccc}0& 0& 0& 0& \cdots & 0\\ {\mathrm{\alpha }}_1^0& {\mathrm{\alpha }}_1^0& {\mathrm{\alpha }}_1^0& {\mathrm{\alpha }}_1^0& \cdots & {\mathrm{\alpha }}_1^0\\ {\mathrm{\alpha }}_1^1& {\mathrm{\alpha }}_1^1+{\mathrm{\alpha }}_2^0& {\mathrm{\alpha }}_1^1+{\mathrm{\alpha }}_2^0& {\mathrm{\alpha }}_1^1+{\mathrm{\alpha }}_2^0& \cdots & {\mathrm{\alpha }}_1^1+{\mathrm{\alpha }}_2^0\\ {\mathrm{\alpha }}_1^2& {\mathrm{\alpha }}_1^2+{\mathrm{\alpha }}_2^1& {\mathrm{\alpha }}_1^2+{\mathrm{\alpha }}_2^1+{\mathrm{\alpha }}_3^0& {\mathrm{\alpha }}_1^2+{\mathrm{\alpha }}_2^1+{\mathrm{\alpha }}_3^0& \cdots & {\mathrm{\alpha }}_1^2+{\mathrm{\alpha }}_2^1+{\mathrm{\alpha }}_3^0\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ {\mathrm{\alpha }}_1^{\mathrm{n}-2}& {\mathrm{\alpha }}_1^{\mathrm{n}-2}+{\mathrm{\alpha }}_2^{\mathrm{n}-3}& \sum _{\mathrm{i}=1}^3{\mathrm{\alpha }}_{\mathrm{i}}^{\mathrm{n}-\mathrm{i}-1}& \sum _{\mathrm{i}=1}^4{\mathrm{\alpha }}_{\mathrm{i}}^{\mathrm{n}-\mathrm{i}-1}& \cdots & \sum _{\mathrm{i}=1}^{\mathrm{j}}{\mathrm{\alpha }}_{\mathrm{i}}^{\mathrm{n}-\mathrm{i}-1}\end{array}\right] \end{gathered}$$

Seja ${\mathrm{S}}_{\mathrm{i}}$, com i variando de 1 a n, a soma dos elementos da i-ésima coluna de A.

$$\begin{gathered} {\mathrm{S}}_1=\mathrm{n}-1 \\ {\mathrm{S}}_1+{\mathrm{S}}_2=\mathrm{n}-2 \\ {\mathrm{S}}_1+{\mathrm{S}}_2+{\mathrm{S}}_3=\mathrm{n}-3 \end{gathered}$$

E assim por diante até:

$\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{S}}_{\mathrm{i}}=\mathrm{n}-\mathrm{n}=0$

No entanto, ao subtrair as equações duas a duas, obtém-se ${\mathrm{S}}_{\mathrm{i}}=-1$, para todo i > 1, um resultado absurdo, pois ${\mathrm{S}}_{\mathrm{n}}=0$ é formada apenas por zeros. Conclui-se, portanto, que não existem tais matrizes descritas no problema.