$$|A-x I|=0$$

Roots are $$-$$1 and 3

Sum of roots $$=\operatorname{tr}(A)=2$$

Product of roots $$=|\mathrm{A}|=-3$$

Let $$A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$$

We have $$\mathrm{a}+\mathrm{d}=2$$

$$\mathrm{ad}-\mathrm{bc}=-3$$

$$A^2=\left[\begin{array}{ll}a & b \\ c & d \end{array}\right] \times\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{ll} a^2+b c & a b+b d \\ a c+c d & b c+d^2 \end{array}\right]$$

We need $$a^2+b c+b c+d^2$$

$$\begin{aligned} & =a^2+2 b c+d^2 \\ & =(a+d)^2-2 a d+2 b c \\ & =4-2(a d-b c) \\ & =4-2(-3) \\ & =4+6 \\ & =10 \end{aligned}$$