Let $A = \left( {\matrix{ 2 & { - 1} \cr 0 & 2 \cr } } \right)$. If $$B = I - {}^5{C_1}(adj\,A) + {}^5{C_2}{(adj\,A)^2} - \,\,.....\,\, - {}^5{C_5}{(adj\,A)^5}$$, then the sum of all elements of the matrix B is

<p>Given $A = \left[ {\matrix{ 2 & { - 1} \cr 0 & 2 \cr } } \right]$</p> <p>and</p> <p>$$B = I - {5_{{C_1}}}(adj\,A) + {5_{{C_2}}}{(adj\,A)^2} - {5_{{C_3}}}{(adj\,A)^3} + {5_{{C_4}}}{(adj\,A)^4} - {5_{{C_5}}}{(adj\,A)^5}$$</p> <p>$= {\left( {I - (adj\,A)} \right)^5}$</p> <p>Cofactor of $$A = \left[ {\matrix{ {{{( - 1)}^{1 + 1}}\,.\,2} & {{{( - 1)}^{1 + 2}}\,.\,0} \cr {{{( - 1)}^{2 + 1}}\,.\,( - 1)} & {{{( - 1)}^{2 + 2}}\,.\,2} \cr } } \right]$$</p> <p>$= \left[ {\matrix{ 2 & 0 \cr 1 & 2 \cr } } \right]$</p> <p>Transpose of cofactor of $A = \left[ {\matrix{ 2 & 1 \cr 0 & 2 \cr } } \right]$</p> <p>$\therefore$ $adj\,A = \left[ {\matrix{ 2 & 1 \cr 0 & 2 \cr } } \right]$</p> <p>Now, $I - adj\,A$</p> <p>$$ = \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right] - \left[ {\matrix{ 2 & 1 \cr 0 & 2 \cr } } \right]$$</p> <p>$= \left[ {\matrix{ { - 1} & { - 1} \cr 0 & { - 1} \cr } } \right]$</p> <p>Now let,</p> <p>$$P = I - adj\,A = \left[ {\matrix{ { - 1} & { - 1} \cr 0 & { - 1} \cr } } \right]$$</p> <p>$\therefore$ $${P^2} = \left[ {\matrix{ { - 1} & { - 1} \cr 0 & { - 1} \cr } } \right]\left[ {\matrix{ { - 1} & { - 1} \cr 0 & { - 1} \cr } } \right]$$</p> <p>$= \left[ {\matrix{ 1 & 2 \cr 0 & 1 \cr } } \right]$</p> <p>$${P^4} = {P^2}\,.\,{P^2} = \left[ {\matrix{ 1 & 2 \cr 0 & 1 \cr } } \right]\left[ {\matrix{ 1 & 2 \cr 0 & 1 \cr } } \right] = \left[ {\matrix{ 1 & 4 \cr 0 & 1 \cr } } \right]$$</p> <p>$${P^5} = {P^4}\,.\,P = \left[ {\matrix{ 1 & 4 \cr 0 & 1 \cr } } \right]\left[ {\matrix{ { - 1} & { - 1} \cr 0 & { - 1} \cr } } \right] = \left[ {\matrix{ { - 1} & { - 5} \cr 0 & { - 1} \cr } } \right]$$</p> <p>$\therefore$ $B = \left[ {\matrix{ { - 1} & { - 5} \cr 0 & { - 1} \cr } } \right]$</p> <p>Now sum of elements $= - 1 - 5 - 1 + 0 = - 7$</p>