If AB = A and BA = B, then which of the following is/are true?

Given, AB = A, BA = B

$\Rightarrow \mathrm{B}\times \mathrm{AB}=\mathrm{B}\times \mathrm{A}$

or (BA)B = B or B² = B

Also,

$\begin{array}{r}\mathrm{A}\times \mathrm{BA}=\mathrm{AB}\\ \Rightarrow \left(\mathrm{AB}\right)\mathrm{A}=\mathrm{A}\\ \Rightarrow {\mathrm{A}}^2=\mathrm{A}\end{array}$

Now ${\left({\mathrm{A}}^{\mathrm{T}}\right)}^2=\left({\mathrm{A}}^{\mathrm{T}}\times {\mathrm{A}}^{\mathrm{T}}\right)=(\mathrm{A}\times \mathrm{A}{)}^{\mathrm{T}}={\left({\mathrm{A}}^2\right)}^{\mathrm{T}}={\mathrm{A}}^{\mathrm{T}}$

Similarly, (B^T)² = B^T

Hence, A^T and B^T are idempotent.