Determinants Matrices Question 151
Question: If Z is an idempotent matrix, then $ {{(I+Z)}^{n}} $
Options:
A) $ I+2^{n}Z $
B) $ I+(2^{n}-1)Z $
C) $ I-(2^{n}-1)Z $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
- [b] Z is idempotent then $ Z^{2}=Z\Rightarrow Z^{3},Z^{4},…,Z^{n}=Z $
$ {{(I+Z)}^{n}}={{}^{n}}C_0I^{n}+{{}^{n}}C_1{I^{n-1}}Z+{{}^{n}}C_2{I^{n-2}}Z^{2}+…+{{}^{n}}C _{n}Z^{n} $
$ ={{}^{n}}C_0I+{{}^{n}}C_1Z+{{}^{n}}C_2Z+{{}^{n}}C_3Z+….+{{}^{n}}C _{n}Z $
$ =I+{{(}^{n}}C_1+{{}^{n}}C_2+{{}^{n}}C_3+…+{{}^{n}}C _{n})Z=I+(2^{n}-1)Z $
BETA
