Let A = ( cases 2 & - 1 0 & 2 cases ) . If B = I - C1 5 (adj A) +...

Simplifying the given expression we get,$B = I - C1 5 adj ( A ) + C2 5 ( adj ( A ) {)}^{2} - C3 5 ( adj ( A ) {)}^{3} + C4 5 ( adj ( A ) {)}^{4} - C5 5 ( adj ( A ) {)}^{5}$

$⇒ B = ( I - adj ( A ) {)}^{5}$

Now, A = \left[\begin{cases}2 & - 1 \\ 0 & 2\end{cases}\right] ⇒ adj \left(A\right) = \left[\begin{cases}2 & 1 \\ 0 & 2\end{cases}\right]

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

⇒ - B = {\left[\begin{cases}1 & 1 \\ 0 & 1\end{cases}\right]}^{5} = \left[\begin{cases}1 & 5 \\ 0 & 1\end{cases}\right]

⇒ B = \left[\begin{cases}- 1 & - 5 \\ 0 & - 1\end{cases}\right]

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