I agree that Gandalf's solution is much, much more complicated than the question seems to warrant. In fact, he seems to be writing his own textbook!

However, I do think you need to also prove, in your solution, that positive definite matrices are invertible. This isn't hard, however.

A matrix is a linear mapping

.

It's invertible provided that this mapping is 1-1.

Therefore, to show an inverse of a matrix

M exists, you need to show there are no two (non-equal) vectors

x,

y such that

Mx=

My.

This is equivalent to showing that there is no vector

z (=

x-

y) such that

Mz=0.

But we know this is true, because "positive definite" implies

zMz>0.

Therefore, all positive-definite matrices are invertible!