Let A = [{1, 2}, {0, 1}] and B = I + adj(A) + (adj A)^2 + ...... + (adj A)^{10}. Then the sum of all the elements of the matrix B is

Question

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

(1) -110

(2) 22

(3) -88

(4) -124

Answer 1

Correct option is (3) -88

\(\operatorname{Adj}(A)=\left[\begin{array}{cc}1 & -2 \\ 0 & 1\end{array}\right]\)

\((\operatorname{Adj} A)^{2}=\left[\begin{array}{cc}1 & -4 \\ 0 & 1\end{array}\right]\)

\((\operatorname{Adj} A)^{10}=\left[\begin{array}{cc}1 & -20 \\ 0 & 1\end{array}\right]\)

\(\mathrm{B}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]+\left[\begin{array}{cc}1 & -2 \\ 0 & 1\end{array}\right]+\left[\begin{array}{cc}1 & -4 \\ 0 & 1\end{array}\right]+\ldots+\left[\begin{array}{cc}1 & -20 \\ 0 & 1\end{array}\right]\)

\(\mathrm B=\left[\begin{array}{cc}11 & -110 \\ 0 & 11\end{array}\right] \) 

\(\Rightarrow\) sum of elements of B \(=-88\)