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!
