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

`B=P^(-1) AP`
`implies B^(2)=(P^(-1) AP) (P^(-1) AP)=P^(-1) A^(2)P`
`implies B^(n)=P^(-1) A^(n)P AA n in N`
`|B|=|P^(-1) AP|=|P^(-1)||A||P|=|A|`
`:.` If `|A| ne 0` then `|B| ne 0`
`P^(-1) (A-lambda I)P=P^(-1) AP-P^(-1) lambdaP=B-lambdaI`