Is this a valid proof to prove AB=BA matrices? Can you explain me where I'm wrong please!

You were told what was wrong. (Actually, I'm not sure I see anything here that is right.) You're assuming the desired conclusion.

Can you at least explain your conclusion? How does [MATH]\mathbf{AB} = \mathbf{BA} \implies (\mathbf{AB})^T = (\mathbf{BA})^T[/MATH] let you conclude that [MATH]\mathbf{AB} = \mathbf{BA}[/MATH]?
 
AB = (AB)T=BTAT= BA.

After reading your proof I had to just give you this proof.

(It is funny that as recently as yesterday someone asked me for a proof of this fact)
 
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