Question

Let p be a nonsingular matrix, and $I+p+p^2+.....+p^n=0$, then find $p^{-1}$.

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

Answer 1

The correct option is C $p^n$

We have $I+p+p^2+.....+p^n=O$ ----------$\left(1\right)$
Since p is a nonsingular matrix, p is invertible.
Multiplying both sides of (1) by $p^{-1}$, we get
$p^{-1}+I+Ip+.....+p^{n-1}I=O.p^{-1}$
or $p^{-1}+I(1+p+......+p^{n-1})=O$
or $p^{-1}=-I(I+p+p^2+.....+p^{n-1})=-I(-p^n)=p^n$