Matrices & Determinants
Sum of Matrix Power Series
nta_pyq_2023_apr
Grade 12
Question:
Let $A=\begin{pmatrix}1&5/1\\0&1\end{pmatrix}$, $B=\begin{pmatrix}1&2\\-1&-1\end{pmatrix}A\begin{pmatrix}-1&-2\\1&1\end{pmatrix}$, then the sum of all entries of $\displaystyle\sum_{n=1}^{50}B^n$ is equal to
Step-by-Step Solution
Key Concept: $B=MAN$ where $MN=I$. $B^n=MA^nN$. $A^n=I+nE$ where $E=\begin{pmatrix}0&1/51\\0&0\end{pmatrix}$.
$\sum_{n=1}^{50}B^n$ has entry sum $=100$.
Correct Answer: 4