Question

An invertible \(n \times n\) matrix \(P\) satisfies \(P^3 = I\), where \(I\) is the identity matrix of size \(n\). Which one of the following options is true?

• A. \(P^5 = P\)
• B. \(P^5 = P^{-1}\)
• C. \(P^4 = P^{-1}\)
• D. \(P^4 = P^2\)

Hint:

Use \(P^3=I\) to write \(P^{-1}=P^2\), then reduce higher powers of \(P\) using \(P^3=I\).

Answer 1

The Correct Option is B

Step 1: Use the given relation to express \(P^{-1}\) in terms of \(P\).
We are given \(P^3 = I\), where \(P\) is invertible.
Multiply both sides on the right by \(P^{-1}\):
\[ P^3 P^{-1} = I \cdot P^{-1} \]
\[ P^2 = P^{-1} \]
This is a key relation used to check every option.

Step 2: Express higher powers of \(P\) using \(P^3=I\).
Since \(P^3=I\), any higher power of \(P\) can be reduced by pulling out factors of \(P^3\).
\[ P^4 = P^3 \cdot P = I \cdot P = P \]
\[ P^5 = P^3 \cdot P^2 = I \cdot P^2 = P^2 \]

Step 3: Combine Step 1 and Step 2 to test option (B).
From Step 2, \(P^5 = P^2\). From Step 1, \(P^2 = P^{-1}\).
So \(P^5 = P^{-1}\), which is exactly option (B).

Step 4: Rule out the remaining options using the same relations.
Option (A) \(P^5=P\): since \(P^5=P^2\), this needs \(P^2=P\), which is not implied by \(P^3=I\) in general (it would force \(P=I\)). Not generally true.
Option (C) \(P^4=P^{-1}\): since \(P^4=P\), this needs \(P=P^{-1}\), that is \(P^2=I\); combined with \(P^2=P^{-1}\) from Step 1, this would force \(P^{-1}=I\), so \(P=I\), not general.
Option (D) \(P^4=P^2\): since \(P^4=P\), this needs \(P=P^2\), again forcing \(P=I\). Not generally true.

Final Answer:
The relation that always holds is \(P^5=P^{-1}\).
\[ \boxed{P^5 = P^{-1}} \]