Matrices & Determinants
Nilpotent Matrix — Sum of Elements of B^100
nta_pyq_2026_jan
Grade 12
Question:
Let $A=\begin{bmatrix}3&-4\\1&-1\end{bmatrix}$ and $B$ be two matrices such that $A^{100}=100B+I$. Then the sum of all the elements of $B^{100}$ is _____.
Step-by-Step Solution
Key Concept: Write $A=I+M$ where $M=\begin{bmatrix}2&-4\\1&-2\end{bmatrix}$. $M^2=\begin{bmatrix}0&0\\0&0\end{bmatrix}$, so $M$ is nilpotent. $A^{100}=(I+M)^{100}=I+100M$. Thus $100B=100M\Rightarrow B=M$.
$B^{100}=0$. Sum $=0$.
Correct Answer: 0