Question
If $$A$$ and $$B$$ are two matrices such that $$AB = A$$ and $$BA = B,$$ then which one of the following is correct ?
A.
$${\left( {{A^T}} \right)^2} = {A^T}$$
B.
$${\left( {{A^T}} \right)^2} = {B^T}$$
C.
$${\left( {{A^T}} \right)^2} = {\left( {{A^{ - 1}}} \right)^{ - 1}}$$
D.
None of the above
Answer :
$${\left( {{A^T}} \right)^2} = {A^T}$$
Solution :
Let $$A$$ and $$B$$ be two matrices such that $$AB = A$$ and $$BA = B$$
Now, consider $$AB = A$$
Take Transpose on both side
$$\eqalign{
& {\left( {AB} \right)^T} = {A^T} \cr
& \Rightarrow {A^T} = {B^T}.{A^T}\,\,\,\,.....\left( 1 \right) \cr
& {\text{Now, }}BA = B \cr} $$
Take, Transpose on both side
$$\eqalign{
& {\left( {BA} \right)^T} = {B^T} \cr
& \Rightarrow {B^T} = {A^T}.{B^T}\,\,\,\,.....\left( 2 \right) \cr} $$
Now, from equation (1) and (2). we have
$$\eqalign{
& {A^T} = \left( {{A^T}.{B^T}} \right){A^T} \cr
& {A^T} = {A^T}\left( {{B^T}{A^T}} \right) \cr
& = {A^T}{\left( {AB} \right)^T}\,\,\,\left( {\because {{\left( {AB} \right)}^T} = {B^T} = {B^T}{A^T}} \right) \cr
& = {A^T}.{A^T} \cr
& {\text{Thus, }}{A^T} = {\left( {{A^T}} \right)^2} \cr} $$