Question

Let $P$ be an $m\times m$ matrix such that $P^2=P$. Then $(I+P{)}^n$ equals.

• A. $I+P$
• B. $I+nP$
• C. $I+2^{n}P$
• D. $I+(2^n-1)P$

Answer 1

The correct option is D $I+(2^n-1)P$

Given $P^2=P$

For $n=2,$ ${(P+I)}^n$

${(P+I)}^2=P^2+I^2+2PI{(P+I)}^2=P+I+2PI{(P+I)}^2=I+3PI=I+(2^2-1)PI$

For $n=3$

${(P+I)}^3=P^3+I^3+3P{I}^2+3P^{2}I{(P+I)}^3=(P^2\times P)+I+3PI+3PI{(P+I)}^3=(P\times P)+I+6P1{(P+I)}^3=P^2+I+6PI{(P+I)}^3=P+I+6PI{(P+I)}^3=PI+I+6PI{(P+I)}^3=I+7P1{(P+I)}^3=I+(2^3-1)PI$

So the genreral term is $(P+I{)}^n=I+(2^n-1)PI$

Option $D$ is correct