If $A$ is $3\times 3$ square matrix whose characteristic polynomial equations is ${\lambda }^3-3{\lambda }^2+4=0,$ then trace of $\mathrm{adj}A$ is

If $a, b, c$ are nonzero, then the system of equations, $(\alpha +a)x+\alpha y+\alpha z=0$, $\alpha x+(\alpha +b)y+\alpha z=0$ and $\alpha x+\alpha y+(\alpha +c)z=0$ has a non-trivial solution, if

The value of $a$ for which system of equation $a^3\mathrm{x}+(a+1{)}^3\mathrm{y}+(a+2{)}^3\mathrm{z}=0$, $ax+\left(a+1\right)y+\left(a+2\right)z=0$ and $x+y+z=0$ has non-zero solutions is

$A_1=\left[a_1\right]$

$A_2\mathit{=}\mathit{\left[}\begin{array}{cc}{a}_2& a_3\\ a_4& a_5\end{array}\mathit{\right]}$

$A_3\mathit{=}\left[\begin{array}{lll}{a}_6& a_7& a_8\\ a_9& a_{10}& a_{11}\\ a_{12}& a_{13}& a_{14}\end{array}\right]\mathit{\dots }\mathit{\dots }\mathit{\dots }\mathit{\dots }\mathit{\dots }\mathit{\dots }\mathit{\dots }{A}_n\mathit{=}[\dots \dots .]$

Where $a_r=\mathit{\left[}{\mathit{log}}_{\mathit{2}}\mathit{r}\mathit{\right]}$ ($[.]$ denotes the greatest integer). Then trace of $A_{10}$ is