Proving Symmetry of AAT: Is it Possible?

[KillerZ]

Homework Statement

A matrix S is symmetric if S = S^T. Show that AA^T is symmetric for any matrix A.

Homework Equations

AA^T = (A^T)^TA^T

The Attempt at a Solution

I just said:

A = (A^T)^T and A^T = A^T

Therefore AA^T is symmetric.

I am unsure if that proves it or if I just went in a circle proving nothing.

 

[statdad]

If $$A, B$$ are any matrices, how do you simplify the following expression?

$$\left(AB\right)^T$$

Apply that answer to simplify

$$A A^T$$

and use the hints you provided

 

[KillerZ]

Ok I think this is right.

(AB)^T = B^TA^T

Then (AA^T)^T = AA^T

Therefore AA^T is symmetric because (A^T)^T = A and A^T = A^T.

 

[statdad]

I think you have the correct idea. the 'conventional' way of finishing the proof would be to write everything in one string rather than stopping midstream - I have no idea how
picky your professor would be in that regard.