\hspace{-16}$Determine all Real $\bf{2\times 2}$ matrix $\bf{A=\begin{pmatrix} \bf{a} & \bf{b}\\ \bf{c} & \bf{d} \end{pmatrix}}$ that satisfying \\\\ The equation $\bf{A^2+A+I=O}$
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2 Answers
262A= \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 ??
71Now you need to find all possible values of such a,b,c,d
