If the determinant $\left|\begin{array}{lll}a+p& l+x& u+f\\ b+q& m+y& v+g\\ c+r& n+z& w+h\end{array}\right|$ splits into exactly $\mathrm{K}$ determinants of order $3$, each element of which contains only one term, then find the value of $\mathrm{K}$.

Let $P=\left[\begin{array}{rr}\cos \frac{\mathrm{\pi }}{9}& \sin \frac{\mathrm{\pi }}{9}\\ -\sin \frac{\mathrm{\pi }}{9}& \cos \frac{\mathrm{\pi }}{9}\end{array}\right]$ and $\alpha , \beta , \gamma$ be non-zero real numbers such that $\alpha P^6+\beta P^3+\gamma I$ is the zero matrix. Then find value of ${\left({\alpha }^2+{\beta }^2+{\gamma }^2\right)}^{(\alpha -\beta )(\beta -\gamma )(\gamma -\alpha )}$

Consider an odd order square symmetric matrix $A={\left[a_{ij}\right]}_{n\times n}$. It's elements in any row are $1, 2, \dots \dots , n$ in some order, then prove that $a_{11}, a_{22},\dots \dots \dots .,a_{nn}$ are numbers $1, 2, 3,\dots \dots ,n$ in some order.

Let $A=\left[\begin{array}{lll}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right] ; B=\left[\begin{array}{ccc}2& -1& -1\\ -1& 2& -1\\ -1& -1& 2\end{array}\right]$ and $C=3A+7B$

Prove that

$\left(i\right)$ $(A+B{)}^{2013}=A^{2013}+B^{2013}$
$\left(ii\right)$ $A^n=3^{n-1}A; B^n=3^{n-1}B; C^n=3^{2n-1}A+7.{21}^{n-1}B, n\in N$