Questão 38 - Unicamp - T e X - Unicamp 2027

A $\mathrm{PA} (\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d})$de razão $\mathrm{r}>0$ pode ser reescrita como $(\mathrm{a}, \mathrm{a}+\mathrm{r} , \mathrm{a}+2\mathrm{r}, \mathrm{a}+3\mathrm{r})$.

Do enunciado, tem-se $\mathrm{detM}=- 8\Rightarrow \left|\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right|=-8\Rightarrow \mathrm{ad}-\mathrm{bc}=-8$.

Substituindo $\mathrm{b}=\mathrm{a}+\mathrm{r}, \mathrm{c}= \mathrm{a} +2\mathrm{r} \mathrm{e} \mathrm{d}=\mathrm{a}+3\mathrm{r}$:

$$\begin{gathered} \mathrm{a} (\mathrm{a}+3\mathrm{r}) - (\mathrm{a}+\mathrm{r}) (\mathrm{a}+2\mathrm{r}) = -8 \\ \mathrm{a}²+3\mathrm{ar}-(\mathrm{a}²+3\mathrm{ar}+2\mathrm{r}²)= -8 \\ -2\mathrm{r}²= -8 \\ \mathrm{r}² = 4 \\ \mathrm{r}= \pm 2 \end{gathered}$$

Como $\mathrm{r}<0\Rightarrow \mathrm{r}=-2$

Assim, tem-se a $\mathrm{PA} (\mathrm{a}, \mathrm{a}-2, \mathrm{a}-4, \mathrm{a}-6).$

Ainda do enunciado, tem-se $\mathrm{Tr} \mathrm{M}= -4 \Rightarrow \mathrm{a}+\mathrm{d}=-4.$

Substituindo $\mathrm{d} = \mathrm{a} - 6$:

$\mathrm{a}+\mathrm{a}-6 = -4 \Rightarrow 2\mathrm{a}=2 \Rightarrow \boxed{\mathrm{a}=1}$