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 $\operatorname{tr}(A)=\operatorname{trace}$ of matrix $A, \operatorname{det}(A)=\operatorname{determinant}$ value of matrix $A$ ]

• A. $p =-\operatorname{tr}( A )$
• B. $q =\frac{(\operatorname{tr}( A ))^2-\operatorname{tr}\left( A ^2\right)}{2}$
• C. $r=\operatorname{det}( A )$
• D. $A ^{-1}=\frac{-2\left( A -\operatorname{tr}( A ) I _3\right)}{(\operatorname{tr}( A ))^2-\operatorname{tr}\left( A ^2\right)}$

Answer 1

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

$A=\begin{bmatrix}a&b&c\\ d&e&f\\ g&h&i\end{bmatrix}$
$\left|A-\lambda\right|=0\Rightarrow\begin{vmatrix}a-\lambda&b&c\\ d&e-\lambda&f\\ g&h&i-\lambda\end{vmatrix}=0$
$\begin{array}{l}\Rightarrow-\lambda^3+\lambda^2(a+e+i)-\lambda(a e+e i+i a-g c-h f-b d)+ \\ |A|=0\end{array}$
Using Cayley Hamilton,
$A ^3- A ^2 \operatorname{tr}( A )+\frac{(\operatorname{tr} A)^2-\operatorname{tr}^2}{2} A -| A | . I _3=0$