Question

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

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

Answer 1

Answer: (B)

$I+(2^n-1)Z$

Z is idempotent , then

$Z^2=Z{Z}^3=Z.....Z^n=Z$

$\therefore (I+Z{)}^n={}^n{C}_0{I}^n{+}^n{C}_1{I}^{n-1}Z{+}^n{C}_2{I}^{n-2}{Z}^2+.....{+}^n{C}_n{Z}^n$

$={}^n{C}_{0}I{+}^n{C}_{1}Z{+}^n{C}_{2}Z{+}^n{C}_{3}Z+......{+}^n{C}_{n}Z$

$=I+{(}^n{C}_1{+}^n{C}_2{+}^n{C}_3+......{+}^n{C}_n)Z$

$=I+(2^n-1)Z$