Given a matrix $A$, show that it is positive. 
Show that
  $$A := \begin{bmatrix}7 & 2 & -4\\2 & 4 & -2\\-4 & -2 & 7 \end{bmatrix}$$
  is positive definite.

Could this be proven by showing that each of the vectors of the standard basis gives a positive result, e.g.:
$$\begin{bmatrix}1 & 0 & 0\end{bmatrix} \begin{bmatrix}7 & 2 & -4\\2 & 4 & -2\\-4 & -2  & 7 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \\ 0\end{bmatrix} > 0.$$
The second part of the question asks me to diagonalize the matrix using an orthogonal matrix, which as I understand, is to use elementary matrices on the rows and columns of the matrix to get it to a diagonal form. Would it make a difference if Ifirstly only dealt with the rows and only afterward used the exact matrices only on the columns?
Thanks for your time.

It's easyiest to just find the eigenvalues of the matrix and show they are positive.
