Let A=[array|c|c| 1 & 2 0 & 1 array] and B=I+adj(A)+(adjA)^2+ +(adjA)^10.

Question

Let $A=[\begin{array}{|c|c|} 1 & 2 \ 0 & 1 \end{array}]$ and $B=I+adj(A)+(adjA{)}^{2}+\ldots +(adjA{)}^{10}$.
Then, the sum of all the elements of the matrix $B$ is:

Accepted Answer

$\begin{gathered} adj(A)=[1 -2 \ 0 1] \ (adjA{)}^{2}=[1 -4 \ 0 1] \ (adjA{)}^{3}=[1 -6 \ 0 1] \ (adjA{)}^{4}=[1 -8 \ 0 1] \ (adjA{)}^{r}=[1 (-2r) \ 0 1] $

$ B=\sum _{r=0}^{10}(adjA{)}^{r}=[[|\begin{array}{cc}\sum _{r=0}^{10}1 \sum _{r=0}^{10}(-2r) \ \sum _{r=0}^{10}(0) \sum _{r=0}^{10}(1)\end{array}]]| \ B=[11 -110 \ 0 11] \end{gathered}$

$=-110+11+11=-88$

Solutions · 1