1. The problem statement, all variables and given/known data The problem:
Attached as TheProblem.jpg.

Solution:

2. Relevant equations x^{T} A x

3. The attempt at a solution
I computed the product and got 9x_{2}^{2} + 4 x_{1}x_{1} + 4x_{1}^{2}.

I'm thinking that I might need to show that that obtained polynomial must be below zero since we want to show that matrix A is not a positive definite matrix but, I don't know where to go from here.

That's right, and you know the squares certainly won't be negative so it comes down to the cross term (which I think you mislabeled) to cause a negative number.

The polynomial you wrote down looks like it came from a 2x2 matrix. The matrix in the problem is 3x3. You should have x_{1}, x_{2}, and x_{3} appearing.

In this particular case, the lack of an ##x_2^2## term gives you the freedom to choose a large value for ##x_2## to get the overall sign you want. That's the easiest way to deal with this problem.

In the general case, one way would be to find linear combinations ##y_i## of the ##x_j##'s so that you can write the polynomial in the form ##\lambda_1 y_1^2+ \lambda_2 y_2^2 + \lambda_3 y_3^2##. Then from the signs of the coefficients, you can determine whether the matrix is positive definite or not.

For example, suppose you have the polynomial ##4x_1^2 + 20x_1x_2 + 4x_2^2##. If you let ##y_1 = x_1 + x_2## and ##y_2 = x_1 - x_2##, you'd have
$$4x_1^2 + 20x_1x_2 + 4x_2^2 = 7y_1^2 - 3y_2^2.$$ From the second form, it's clear the polynomial will be negative when ##y_1=0## and ##y_2=1##.

The linear combinations correspond to the eigenvectors of the matrix, and the coefficients, to the eigenvalues. So finding the eigenvalues of the matrix is enough to tell you whether it's positive definite or not.

Is that how this problem is asking to be answered or is that just a neat trick? I'm asking because I don't want to miss anything that's more standard, if it exists.

Finding the eigenvalues? It's a standard way of determining whether a matrix is positive-definite. For this particular problem, however, I think you're just supposed to look at the polynomial and reason out how you can make it negative with the correct choice of ##x_1##, ##x_2##, and ##x_3##.

So, is there any significance to the particular answers given in the solution or do I just need to show that the polynomial is able to, at least, reach a negative value?

The problem wants me to show that the definition of what makes a matrix positive definite fails. I believe what I did accomplishes this but, could you please confirm?