Determinants Matrices Question 79
Question: If B, C are square matrices of order n and if $ A=B+C,BC=CB,C^{2}=0 $ , then for any positive integer $ N,{A^{N+1}}=B^{K}[B+(N+1)C], $ then K/N is
Options:
A) 1
B) ½
C) 2
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
- [a] We have, BC = CB, and $ {A^{N+1}}={{(B+C)}^{N+1}} $
$ {{=}^{N+1}}C_0{B^{N+1}}{{+}^{N+1}}C_1B^{N}C{{+}^{N+1}}C_2{B^{N-1}}C^{2}+ $
$ ….+{{}^{N+1}}C _{r}{B^{N+1-r}}C _{r}+….. $ But given that $ C^{2}=0\Rightarrow C^{3}=C^{4}=…=C^{r}=0 $ Hence, $ {A^{N+1}}={{}^{N+1}}C _{N}{B^{N+1}}+{{}^{N+1}}C_1B^{N}C $
$ ={B^{N+1}}+(N+1)B^{N}C=B^{N}[B+(N+1)C] $ Thus $ K=N $
BETA
