Note
- The sum of two positive definite matrices is positive definite. $(x^T(A+B)x = x^TAx + x^TBx > 0)$.
$i)$ $A+B$ is positive definite from above property.
$ii)$ $x^T(ABA)x = (x^TA)B(Ax) = (Ax)^TB(Ax) > 0$
$iii)$ Since $A$ is symmetric $A^2$ is also symmetric. And $A^2 + I$ is also positive definite since $I$ is positive definite.
Thus, all of the above matrices are positive definite.