Let $A = \begin{pmatrix} 2 & -1 \\ 0 & 2 \end{pmatrix}$. If $B = I - C_1^5 \, \text{adj} \, A + C_2^5 \,…
Let $A = \begin{pmatrix} 2 & -1 \\ 0 & 2 \end{pmatrix}$. If $B = I - C_1^5 \, \text{adj} \, A + C_2^5 \, (\text{adj} \, A)^2 - \ldots - C_5^5 \, (\text{adj} \, A)^5$, then the sum of all elements of the matrix $B$ is:
Solution
Simplifying the given expression we get, $B = I - C_{1}^{5} \text{adj}(A) + C_{2}^{5} (\text{adj}(A))^2 - C_{3}^{5} (\text{adj}(A))^3 + C_{4}^{5} (\text{adj}(A))^4 - C_{5}^{5} (\text{adj}(A))^5$.
$\Rightarrow B = (I - \text{adj}(A))^5$.
Now, $A = \begin{bmatrix} 2 & -1 \\ 0 & 2 \end{bmatrix} \Rightarrow \text{adj}(A) = \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}$.
So, $B = \begin{bmatrix} -1 & -1 \\ 0 & -1 \end{bmatrix}^5$.
$\Rightarrow -B = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}^5 = \begin{bmatrix} 1 & 5 \\ 0 & 1 \end{bmatrix}$.
$\Rightarrow B = \begin{bmatrix} -1 & -5 \\ 0 & -1 \end{bmatrix}$.
Now, sum of elements of matrix $B = -1 -5 -1 = -7$.