Let P be a non - singular matrix such that `I+P+P^(2)+…….P^(n)=O` (where O denotes the null matrix), then `P^(-1)` is

Options

A. `P^(n)`
B. `-P^(n)`
C. `-(1+P+P^(2)+……+P^(n))`
D. None of these

Answer

To solve the problem, we need to analyze the given equation and derive the expression for \( P^{-1} \).

### Step-by-Step Solution:

1. **Given Equation**: We start with the equation:
\[ I + P + P^2 + \ldots + P^n = O \]
where \( O \) is the null matrix.

2. **Rearranging the Equation**: We can rearrange this equation to isolate the identity matrix:
\[ P + P^2 + \ldots + P^n = -I \]

3. **Factoring Out \( P \)**: Notice that we can factor out \( P \) from the left-hand side:
\[ P(I + P + P^2 + \ldots + P^{n-1}) = -I \]

4. **Identifying the Series**: The expression \( I + P + P^2 + \ldots + P^{n-1} \) is a geometric series. The sum of a geometric series can be expressed as:
\[ S = I + P + P^2 + \ldots + P^{n-1} = \frac{I(P^n - I)}{P - I} \]
However, we will use the simpler form for our purposes.

5. **Substituting Back**: We can substitute back into our equation:
\[ P S = -I \]
where \( S = I + P + P^2 + \ldots + P^{n-1} \).

6. **Finding \( P^{-1} \)**: Now we can express \( S \) in terms of \( P^{-1} \):
\[ S = -P^{-1} \]
Thus, we have:
\[ P S = -I \implies P(-P^{-1}) = -I \]
which simplifies to:
\[ -I = -I \]
confirming our derivation.

7. **Final Expression for \( P^{-1} \)**: From our earlier steps, we have:
\[ P^n = -P^{-1} \]
Therefore, we can conclude that:
\[ P^{-1} = -P^n \]

### Conclusion:
Thus, the value of \( P^{-1} \) is:
\[ P^{-1} = -P^n \]