Question

If $Z$ is an idempotent matrix, then $(I+Z{)}^n$ is equal to

• A. $I+2^{n}Z$
• B. $I+(2^n-1)Z$
• C. $I-(2^n-1)Z$
• D. None of the above

Answer 1

The correct option is B $I+(2^n-1)Z$

$Z$ is idempotent, then $Z^2=Z\Rightarrow Z^3=Z^4=\cdots =Z^n=Z\therefore (I+Z{)}^n={}^n{C}_0{I}^n+{}^n{C}_1{I}^{n-1}Z+{}^n{C}_2{I}^{n-2}{Z}^2+\cdots +{}^n{C}_n{Z}^n={}^n{C}_{0}I+{}^n{C}_{1}Z+{}^n{C}_{2}Z+{}^n{C}_{3}Z+\cdots +{}^n{C}_{n}Z=I+({}^n{C}_1+{}^n{C}_2+{}^n{C}_3+\cdots +{}^n{C}_n)Z$ $=I+(2^n-1)Z$