Symbolic Real Positive Definite Matrix

I’d like to define a symbolic real positive definite matrix. For the 2 x 2 example, I thought I could define four real variables using $Assumptions = {a,b,c,d} esc elem esc Reals. And then define the positive definite matrix using MatrixPD = Transpose[{{a,b},{c,d}}].{{a,b},{c,d}}].

However PositiveDefiniteMatrixQ[MatrixPD] returns False. Is this because, I have not defined the assumptions sufficiently to guarantee that MatrixPD will be positive definite, or is it because of a limitation of PositiveDefiniteMatrixQ.

I also tried the following. The manaul includes the following symbolic example
PositiveDefiniteMatrixQ[{{1, a}, {-Conjugate[a], 1}}]
Since I had defined “a” to be real, I assumed I could modify this to be
PositiveDefiniteMatrixQ[{{1, a}, {-a, 1}}]
However, this returned False. Am I not asserting the assumption that “a” is real properly or is something else going wrong?