Positive definite Negative definate Matrix

[retspool]

So i need to show if a matrix is positive definite or negative or neither

I have a matrix 3x3 A so i compute xTAx

I am left with an quadratic equation interms of x_1 x_2 and x_3

The Matrix is of the form

A = [ 9 -1 2]
|-1 7 -3|
[2 -3 3]

and solving xTAx i get

9x_1^2 + 7x_2^2 + 3x_3^2 - 2x_1x_2 + 4x_1x_3 - 6x_3x_2

If i need to show that it is positive definite or negative do i need to find an x and set it equal to (x_1 x_2 x_3) st that the xTAx is greater or less than 0?


Also do i need to find eigenvalues to prove anything?

Thanks

 

[retspool]

Here is what i came up with.
Pretty straightftwd

Attachment enclosed