
Symmetric matrix
If $\displaystyle A$ is a square matrix and $\displaystyle x^TAy=(Ax)^Ty$ for all $\displaystyle x,y\in R^n$, then prove $\displaystyle A$ is a symmetric matrix. ie. prove $\displaystyle A^T=A$
I cant just go $\displaystyle x^TAy=x^TA^Ty$ rite? So what should i do instead?

Use the matrix property $\displaystyle (AB)^T=B^TA^T.$

yeh i know that property but how do i do it? ive just used the property and now im at $\displaystyle x^TAy=x^TA^Ty$. But i cant just cancel them out to get $\displaystyle A=A^T$ because they are matrices...

You're told that the relationship is true for all $\displaystyle x$ and $\displaystyle y$, so it's certainly true for specific $\displaystyle x$ and $\displaystyle y.$
Choose $\displaystyle x$ to consist of entirely of zeros apart from a 1 in the ith position and $\displaystyle y$ to consist entirely of zeros apart from 1 in the jth position. Then evaluate both sides of the equation $\displaystyle x^TA^Ty=x^TAy.$

You started with:
A is nxn
x^TAy= (Ax)^Ty
so
x^TAy= x^TA^Ty
x^TAy  x^TA^Ty = 0
factor...
x^T(A  A^T)y = 0
this is true for all x,y in R^n
consider x = 0, and y = anything in R^n we get x^T(A  A^T)y = 0
consider y = 0, and x = anything in R^n we get x^T(A  A^T)y = 0
consider y=x=0 we get x^T(A  A^T)y = 0
we know the statement is true for all x,y not just x, y, being zero vectors
the only way x^T(A  A^T)y = 0 is true for any x and y is if A A^T = zero matrix
hope this helped