Let $A=\left(\begin{array}{cc}2& -1\\ 0& 2\end{array}\right)$. If $B=I-{}^{5}C_1\left(adj A\right)+{}^{5}C_2(adj A{)}^2\mathit{-}\mathit{ }\mathit{.}\mathit{.}\mathit{.}\mathit{-}\mathit{ }{}^{\mathit{5}}C_{\mathit{5}}{(\mathrm{adj} A)}^{\mathit{5}}$, then the sum of all elements of the matrix $B$ is:

Simplifying the given expression we get,$B=I-{}^{5}C_{1}adj\left(A\right)+{}^{5}C_2(adj(A){)}^2-{}^{5}C_3(adj\left(A\right){)}^3+{}^{5}C_4(adj(A){)}^4-{}^{5}C_5(adj\left(A\right){)}^5$

$\Rightarrow B=(I-adj(A){)}^5$

Now, $A=\left[\begin{array}{cc}2& -1\\ 0& 2\end{array}\right]\Rightarrow adj\left(A\right)=\left[\begin{array}{ll}2& 1\\ 0& 2\end{array}\right]$

So,$B={\left[\begin{array}{cc}-1& -1\\ 0& -1\end{array}\right]}^5$

$\Rightarrow -B={\left[\begin{array}{ll}1& 1\\ 0& 1\end{array}\right]}^5=\left[\begin{array}{ll}1& 5\\ 0& 1\end{array}\right]$

$\Rightarrow B=\left[\begin{array}{cc}-1& -5\\ 0& -1\end{array}\right]$

Now, sum of elements of matrix $B$ $=-1-5-1=-7$