JEE Main 2022 (Online) 29th June Evening ShiftQuestion 1
Let $A=\begin{pmatrix}2&-1\\0&2\end{pmatrix}$. If $B=I-{}^{5}C_1(\operatorname{adj}A)+{}^{5}C_2(\operatorname{adj}A)^{2}-\ldots-{}^{5}C_5(\operatorname{adj}A)^{5}$, then the sum of all elements of the matrix $B$ is
Matrices
Solution
$\operatorname{adj}A=\begin{pmatrix}2&1\\0&2\end{pmatrix}=:C$. $B=\sum_{k=0}^{5}\binom5k(-C)^k\cdot$ with the last sign matching, i.e. $B=(I-C)^5=\begin{pmatrix}1&-1\\0&-1\end{pmatrix}^5$... compute: $I-C=\begin{pmatrix}-1&-1\\0&-1\end{pmatrix}=-(I+N)$, $N=\begin{pmatrix}0&1\\0&0\end{pmatrix}$. $B=-(I+N)^5=-(I+5N)=\begin{pmatrix}-1&-5\\0&-1\end{pmatrix}$. Sum of elements $=-7$. Option (C).