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Say $A$ and $B$ are symmetric, positive definite matrices.

  I've proved that $\det(A+B) \ge \det(A) + \det(B)$ in the case that $A$ and $B$ are two dimensional.

  Is this true in general for $n$-dimensional matrices?

  Is it even true that $\det(A+B) \ge \det(A)$ [as this would also be enough..]

Thanks.

Say $A$ and $B$ are symmetric, positive definite matrices. I've proved that

$$\det(A+B) \ge \det(A) + \det(B)$$

in the case that $A$ and $B$ are two dimensional. Is this true in general for $n$-dimensional matrices? Is the following even true?

$$\det(A+B) \ge \det(A)$$

This would also be enough. Thanks.