Prove that diagonal matrices are symmetric matrices

[Mr Davis 97]

Homework Statement

Same as title.

Homework Equations

The Attempt at a Solution

A defining property of a diagonal matrix is that $A_{ij} = A_{ji} ~~\forall i,j \le n$. This means that $((A)^{t})_{ji} = A_{ji}$. Therefore, we know that $A^t = A$. This shows that a diagonal matrix is symmetric.

Is this an okay proof? Am I making too big of a leap in logic to start with $A_{ij} = A_{ji} ~~\forall i,j \le n$? Or do I need to first prove that that statement is true for diagonal matrices?

 

[fresh_42]

I would say, in the case of a diagonal matrix, there is nothing to prove, since all $A_{ij} = 0 = A_{ji}$ for $i \neq j$ and of course is $A_{ii}=A_{ii}$ for the rest.