Determinants Matrices Question 159
Question: If A and B are two matrices such that AB = A and BA=B, then which one of the following is correct ?
Options:
A) $ {{(A^{T})}^{2}}=A^{T} $
B) $ {{(A^{T})}^{2}}=B^{T} $
C) $ {{(A^{T})}^{2}}={{({A^{-1}})}^{-1}} $
D) None of the above
Show Answer
Answer:
Correct Answer: A
Solution:
- [a] Let A and B be two matrices such that AB = A and BA = B Now, consider AB = A Take transpose on both side $ {{(AB)}^{T}}=A^{T} $
$ \Rightarrow A^{T}=B^{T}.A^{T}…(1) $ Now, $ BA=B $ Take, transpose on both side $ {{(BA)}^{T}}=B^{T} $
$ \Rightarrow B^{T}=A^{T}.B^{T}….(2) $ Now, from equation (1) and (2). We have $ A^{T}=(A^{T}.B^{T})A^{T} $
$ A^{T}=A^{T}(B^{T}A^{T}) $
$ =A^{T}{{(AB)}^{T}}(\because {{(AB)}^{T}}=B^{T}=B^{T}A^{T}) $
$ =A^{T}.A^{T} $ Thus, $ A^{T}={{(A^{T})}^{2}} $
BETA
