Question

For a $3\times 3$ invertible matrix $A$ satisfying the characteristic equation $A^3+pA{A}^2+qA-rI=0$, which of the following is/are true [Here $\mathrm{tr}(A)=\mathrm{trace}$ of matrix $A,\det (A)=\mathrm{determinant}$ value of matrix $A$ ]

• A. $p=-\mathrm{tr}(A)$
• B. $q=\frac{(\mathrm{tr}(A){)}^2-\mathrm{tr}\left(A^2\right)}{2}$
• C. $r=\det (A)$
• D. $A^{-1}=\frac{-2\left(A-\mathrm{tr}(A)I_3\right)}{(\mathrm{tr}(A){)}^2-\mathrm{tr}\left(A^2\right)}$

Answer 1

Answer: (A), (B), (C)

$A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$
$\left|A-\lambda \right|=0\Rightarrow \left|\begin{array}{ccc}a-\lambda & b& c\\ d& e-\lambda & f\\ g& h& i-\lambda \end{array}\right|=0$
$\begin{array}{l}\Rightarrow -{\lambda }^3+{\lambda }^2(a+e+i)-\lambda (ae+ei+ia-gc-hf-bd)+\\ |A|=0\end{array}$
Using Cayley Hamilton,
$A^3-A^2\mathrm{tr}(A)+\frac{(\mathrm{tr}A{)}^2-{\mathrm{tr}}^2}{2}A-|A|.I_3=0$