Let $A=\left[\begin{array}{cc}2 & -1 \ 0 & 2\end{array} ight]$. If $\mathrm{B}=\mathrm{I}-{ }^3 \mathrm{C}_1(\operatorname{adj} \mathrm{A})+{ }^3 \mathrm{C}_2(\operatorname{adj} \mathrm{A})^2-{ }^3 \mathrm{C}_3(\operatorname{adj} \mathrm{A})^3$, then the sum of all elements of the matrix $B$ is

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Let $A=\left[\begin{array}{cc}2 & -1 \ 0 & 2\end{array}ight]$. If $\mathrm{B}=\mathrm{I}-{ }^3 \mathrm{C}_1(\operatorname{adj} \mathrm{A})+{ }^3 \mathrm{C}_2(\operatorname{adj} \mathrm{A})^2-{ }^3 \mathrm{C}_3(\operatorname{adj} \mathrm{A})^3$, then the sum of all elements of the matrix $B$ is

Aicharya · Accepted Answer

$\begin{aligned} \mathrm{B} & =\mathrm{I}-{ }^3 \mathrm{C}_1(\operatorname{adj} \mathrm{A})+{ }^3 \mathrm{C}_2(\operatorname{adj} \mathrm{A})^2-{ }^3 \mathrm{C}_3(\operatorname{adj} \mathrm{A})^3 \ & =\mathrm{I}-3 \operatorname{adj} \mathrm{A}+3(\operatorname{adj} \mathrm{A})^2-1(\operatorname{adj} \mathrm{A})^3 \ & =(\mathrm{I}-\operatorname{adj} \mathrm{A})^3 \ \operatorname{adj} \mathrm{A} & =\left[\begin{array}{ll} 2 & 1 \ 0 & 2 \end{array}ight] \ 	herefore \quad \mathrm{B} & =\left(\left[\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}ight]-\left[\begin{array}{ll} 2 & 1 \ 0 & 2 \end{array}ight]ight)^3 \ & =\left[\begin{array}{rr} -1 & -1 \ 0 & -1 \end{array}ight] \ & =\left[\begin{array}{rr} -1 & -1 \ 0 & -1 \end{array}ight]\left[\begin{array}{rr} -1 & -1 \ 0 & -1 \end{array}ight]\left[\begin{array}{cc} -1 & -1 \ 0 & -1 \end{array}ight] \ & =\left[\begin{array}{rr} 1 & 2 \ 0 & 1 \end{array}ight]\left[\begin{array}{rr} -1 & -1 \ 0 & -1 \end{array}ight] \ & =\left[\begin{array}{rr} -1 & -3 \ 0 & -1 \end{array}ight] \end{aligned}$ Sum of all elements of the matrix B $=-1-3-1=-5$

Let $A=\left[\begin{array}{cc}2 & -1 \\ 0 & 2\end{array}\right]$. If $\mathrm{B}=\mathrm{I}-{ }^3…

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