P is a non-singular matrix and A, B are two matrices such that `B=P^(-1) AP`. The true statements among the following are

Options

A. A is invertible iff B is invertib,e
B. `B^(n)=P^(-1) A^(n) P AA n in N`
C. `AA lambda in R, B-lambdaI=P^(-1) (A-lambdaI)P`
D. A and B are both singular matrices

Explanation

To solve the problem, we need to analyze the statements given in the context of the relationship between matrices A, B, and P, where \( B = P^{-1} A P \).

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We are given that \( B = P^{-1} A P \). This means that B is a similarity transformation of A using the non-singular matrix P.

2. **Checking Statement 1**:
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