Proving that $(A^t)^t=A$

Question

$(A^t)^t=A$
$(A^t)^t-A=0$
$(A^t)^t-A=A-A\rightarrow (A^t)^t=A$

Is this proof is valid or do I need to add more information to make it more clear?

Answer 1

This is not a proof. There is no argument, you just bring $A$ to the other side and then use what you're supposed to prove.

Let us denote the $(i,j)$th entry of a matrix $M$ by $M_{ij}$. Two matrices are equal if all their entries are equal.

$A^t$ is defined as the matrix with entries $A_{ji}$.

Thus $$((A^t)^t)_{ij}=(A^t)_{ji}=A_{ij}$$

so $(A^t)^t=A$.