How to check if a matrix is positive definite I want to know how to check if a matrix M is positive definite
,assume that M is 3x3 real numbers matrix

I think one way is to put the matrix in a quadratic form  $X^TMX$
 , where X is a vector $X^T=[x_1   x_2   x_3]$
 , my question is if I found that $X^TMX = ax_1^2 + bx_1*x_2+ ........$ can I say that the matrix M is not positive definite because the term $bx_1*x_2$ can be negative or I have to try to put the value of $X^TMX$ in the form of sum of squares e.g.,$()^2+()^2+.....$ and then decide?

and what is the relation between the positive definiteness of a matrix and its determinant?
I don't think there is a nice answer for matrices in general. Most often we care about positive definite matrices for Hermitian matrices, so a lot is known in this case.

The one I always have in mind is that a Hermitian matrix is positive definite iff its eigenvalues are all positive.

Glancing at the wiki article on this alerted me to something I had not known, Sylvester's criterion which says that you can use determinants to test (a Hermitian matrix) for positive definiteness by checking to see if all the square submatrices whose upper left corner is the $(1,1)$ entry have positive determinant.

Sorry if this is repeating things you already know, but it's the most useful information I can provide. Good luck!
Just do calculation of the term X^TAX and then check whether the answer can be negative or not. if it can be negative then it is not positive definite or vice versa for example if answer comes to be x1^2+x2^2+x3^2 then it can never be negative as there are squared terms so in this case matrix A will be positive definite.