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An $n \times n$ matrix $M$ with real entries is said to be positive definite if for every non-zero $n$-dimensional vector $x$ with real entries, we have $x^{T}Mx>0.$ Let  $A$ and $B$ be symmetric, positive definite matrices of size $n\times n$ with real entries.

Consider the following matrices, where $I$ denotes the $n\times n$ identity matrix:

  1. $A+B$
  2. $ABA$
  3. $A^{2}+I$

Which of the above matrices must be positive definite?

  1. Only $(2)$
  2. Only $(3)$
  3. Only $(1)$ and $(3)$
  4. None of the above matrices are positive definite
  5. All of the above matrices are positive definite
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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. 

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