Martix equation....

A= \begin{pmatrix} a &b \\ c& d \end{pmatrix}

to find characteristic equation of the matrix ,

det ( A - xI) = 0

\begin{vmatrix} a-x &b \\ c &d-x \end{vmatrix} =0

x² - (a+d)x + (ad-bc) = 0

now replacing x with A ,

A² - (a+d)A + (ad-bc)I =0

comparing with given eqn,

A² +A +I =0

we have , a+d =-1 and ad-bc = 1

what next ??

Now you need to find all possible values of such a,b,c,d