CMI-2018-DataScience-A: 1

Question

If $P$ is an invertible matrix and $A=PBP^{-1},$ then which of the following statements are necessarily true?

• A. $B=P^{-1}AP$
• B. $|A|=|B|$
• C. $A$ is invertible if and only if $B$ is invertible
• D. $B^T=A^T$

Answer 1

Pre multiplying and post multiplying both sides by P^-1 and P respectively, we can conclude that B=P^-1AP.

 

Option A is correct

Answer 2

|P^(-1)| = 1/ |P| 

So, |A| = |P|*|B|*|P^(-1)|
            = |B|......{As the determinant of P gets cancelled out}
Hence, OPTION (B.) is also TRUE...!!

As, P is invertible...
So, A will be inverible if and only if the B will be invertible, as |A| = |B|...!!

Hence, OPTION C is also TRUE...!!

But, the option D isn't necessarily TRUE, as |A| = |B| can be satisfied even if the matrices aren't equal...!!
So, in the cases where matrices aren't same, their Transpose won't be same..!!

Hence, answers are.... 
(A.), (B.), (C.)..!!!