Let A=pmatrix2&-10&2pmatrix.

Question

Let $A=\begin{pmatrix}2&-1\0&2\end{pmatrix}$. If $B=I-{}^5C_1(\operatorname{adj}A)+{}^5C_2(\operatorname{adj}A)^2-{}^5C_3(\operatorname{adj}A)^3+{}^5C_4(\operatorname{adj}A)^4-{}^5C_5(\operatorname{adj}A)^5$, then the sum of all elements of the matrix $B$ is

Accepted Answer

$B=(I-\operatorname{adj}A)^5$ (binomial expansion in reverse).

$\operatorname{adj}A=\begin{pmatrix}2&1\0&2\end{pmatrix}$, so $I-\operatorname{adj}A=\begin{pmatrix}-1&-1\0&-1\end{pmatrix}=P$.

$P^2=\begin{pmatrix}1&2\0&1\end{pmatrix}$, $P^4=\begin{pmatrix}1&4\0&1\end{pmatrix}$, $P^5=P^4P=\begin{pmatrix}-1&-5\0&-1\end{pmatrix}$.

So $B=\begin{pmatrix}-1&-5\0&-1\end{pmatrix}$, and the sum of all elements is $-1-5+0-1=-7$.

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