Question

If $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 0& -2& 4\end{array}\right],6A^{-1}=A^2+cA+dI$, then $(c,d)$ is equal to

• A. $(-6,11)$
• B. $(-11,6)$
• C. $(11,6)$
• D. $(6,11)$

Answer 1

The correct option is A $(-6,11)$

Every square matrix $A$ satisfies its characteristic equation i.e $|A-\lambda I|=0$
$\Rightarrow \left|\begin{array}{ccc}1-\lambda & 0& 0\\ 0& 1-\lambda & 1\\ 0& -2& 4-\lambda \end{array}\right|=0$
$\Rightarrow {\lambda }^3-6{\lambda }^2+11{\lambda }^{}-6=0$
$\Rightarrow A^3-6A^2+11A-6I=0\Rightarrow 6I=A^3-6A^2+11A$
$\Rightarrow 6A^{-1}=A^2-6A^{}+11I$
(multiplying both sides by $A^{-1}$)
$\Rightarrow 6A^{-1}=A^2+cA+dI$
$\Rightarrow c=-6;d=11$