Matrices & Determinants
Sum of adj Powers — 2×2 Matrix
nta_pyq_2024_apr
Grade 12
Question:
Let $A=\begin{bmatrix}1&2\\0&1\end{bmatrix}$ and $B=I+\text{adj}(A)+(\text{adj}A)^2+\cdots+(\text{adj}A)^{10}$. Then, the sum of all the elements of the matrix $B$ is:
Step-by-Step Solution
Key Concept: For $A=\begin{bmatrix}1&2\\0&1\end{bmatrix}$, $\text{adj}(A)=\begin{bmatrix}1&-2\\0&1\end{bmatrix}$. Powers: $(\text{adj}A)^k=\begin{bmatrix}1&-2k\\0&1\end{bmatrix}$. Sum $B=\sum_{k=0}^{10}\begin{bmatrix}1&-2k\\0&1\end{bmatrix}$.
$B=\begin{bmatrix}11&-110\\0&11\end{bmatrix}$. Sum $=-88$.
Correct Answer: 3