Question

qual o valor do determinante abaixo?

Answer 1

$\left|\begin{array}{ccc}\text{A}&1&1\\\text{A}&-1&-\text{A}\\\text{A}^2&1&\text{A}\end{array}\right|~\begin{array}{ccc}\text{A}&1\\\text{A}&-1&\\\text{A}^2&1\end{array}$

$\text{Det}=\text{A}\cdot(-1)\cdot\text{A}+1\cdot(-\text{A})\cdot\text{A}^2+1\cdot\text{A}\cdot1-1\cdot(-1)\cdot\text{A}^2-\text{A}\cdot(-\text{A})\cdot1-1\cdot\text{A}\cdot\text{A}$

$\text{Det}=-\text{A}^2-\text{A}^3+\text{A}+\text{A}^2+\text{A}^2-\text{A}^2$

$\text{Det}=-\text{A}^3-2\text{A}^2+2\text{A}^2+\text{A}$

$\boxed{\text{Det}=-\text{A}^3+\text{A}}$

Exemplo: para $\text{A}=2$, temos:

$\left|\begin{array}{ccc}2&1&1\\2&-1&-2\\4&1&2\end{array}\right|$

Calculando o determinante:

$\left|\begin{array}{ccc}2&1&1\\2&-1&-2\\4&1&2\end{array}\right|~\begin{array}{ccc}2&1\\2&-1&\\4&1\end{array}$

$\text{Det}_2=2\cdot(-1)\cdot2+1\cdot(-2)\cdot4+1\cdot2\cdot1-1\cdot(-1)\cdot4+2\cdot(-2)\cdot1+1\cdot2\cdot2$

$\text{Det}_2=-4-8+2+4+4-4$

$\text{Det}_2=-6$

E substituindo $\text{A}$ por $2$ na expressão encontrada, obtemos:

$\text{Det}=-\text{A}^3+\text{A}$

$\text{Det}_2=-2^3+2$

$\text{Det}_2=-8+2$

$\text{Det}_2=-6$, como obtemos anteriormente.