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Yet another way to see this is to note that $A = \overline{Q}^{t}Q$ for some invertible matrix $Q$. Then ${\rm det}(A+B) = |{\rm det}(Q)|^{2}{\rm det}{( I + (\overline{Q}^{-1}})^{t}BQ^{-1})$. Now $(\overline{Q}^{-1})^{t}BQ^{-1}$ is Hermitian, and positive definite. It suffices to prove that if $X$ is positive definite and Hermitian, then ${\rm det}(I+X) \geq (1 + {\rm det}X)$. We may conjugate $X$ by a unitary matrix $U$ and assume that $X$ is diagonal. Let the eigenvalues of $X$ be $\{ \lambda_{1},\ldots, \lambda_{n} \lambda_{1},\ldots, \}$lambda_{n}$, (allowing repetitions). Then${\rm det}(I+X) = \prod_{i=1}^{n}(1 + \lambda_{i}) \geq 1 + \prod_{i=1}^{n} \lambda_{i} = 1 + {\rm det}X.$Such an argument appears in some proofs by R. Brauer, though I do not know whether it originates with him. Later edit: Incidentally, I think that with the arithemetic-geometric mean inequality and a slightly more careful analysis, you can see by this approach that for$X$as above, you do have${\rm det}(I+X) \geq (1 +({\rm det}X)^{1/n})^{n}$(a special case of the inequality of Minkowski mentioned in the accepted answer, but enough to prove the general case by an argument similar to that above). For set$d = {\rm det}X$. Let$s_{m}(\lambda_{1},\ldots ,\lambda_{n})$denote the$m$-th elementary symmetric function evaluated at the eigenvalues. Using the arithmetic-geometric mean inequality yields that$s_{m}(\lambda_{1},\ldots ,\lambda_{n}) \geq \left( \begin{array}{clcr} n\\m \end{array} \right)d^{m/n}$, so we obtain${\rm det}(I+X) \geq (1+d^{1/n})^{n}.$5 Minor textual changes. Yet another way to see this is to note that$A = \overline{Q}^{t}Q$for some invertible matrix$Q$. Then${\rm det}(A+B) = |{\rm det}(Q)|^{2}{\rm det}{( I + (\overline{Q}^{-1}})^{t}BQ^{-1})$. Now$(\overline{Q}^{-1})^{t}BQ^{-1}$is Hermitian, and positive definite. It suffices to prove that if$X$is positive definite and Hermitian, then${\rm det}(I+X) \geq (1 + {\rm det}X)$. We may conjugate$X$by a unitary matrix$U$and assume that$X$is diagonal. Let the eigenvalues of$X$be$\{ \lambda_{1},\ldots, \lambda_{n} \}$(allowing repetitions). Then${\rm det}(I+X) = \prod_{i=1}^{n}(1 + \lambda_{i}) \geq 1 + \prod_{i=1}^{n} \lambda_{i} = 1 + {\rm det}X.$Such an argument appears in some proofs by R. Brauer, though I do not know whether it originates with him. Later edit: Incidentally, I think that with the arithemetic-geometric mean inequality and a slightly more careful analysis, you can see by this approach that for$X$as above, you do have${\rm det}(I+X) \geq (1 +({\rm det}X)^{1/n})^{n}$(a special case of the inequality of Minkowski mentioned in the accepted answer)answer, but enough to prove the general case by an argument similar to that above). For set$d = {\rm det}X$. Let$s_{m}(\lambda_{1},\ldots ,\lambda_{n})$denote the$m$-th elementary symmetric function evaluated at the eigenvalues. Using the arithmetic-geometric mean inequality yields that$s_{m}(\lambda_{1},\ldots ,\lambda_{n}) \geq \left( \begin{array}{clcr} n\\m \end{array} \right)d^{m/n}$, so we obtain${\rm det}(I+X) \geq (1+d^{1/n})^{n}.$4 Minor textual changes. Yet another way to see this is to note that$A = \overline{Q}^{t}Q$for some invertible matrix$Q$. Then${\rm det}(A+B) = |{\rm det}(Q)|^{2}{\rm det}{( I + (\overline{Q}^{-1}})^{t}BQ^{-1})$. Now$(\overline{Q}^{-1})^{t}BQ^{-1}$is Hermitian, and positive definite. It suffices to prove that if$X$is positive definite and Hermitian, then${\rm det}(I+X) \geq (1 + {\rm det}X)$. We may conjugate$X$by a unitary matrix$U$and assume that$X$is diagonal. Let the eigenvalues of$X$be$\{ \lambda_{1},\ldots, \lambda_{n} \}$(allowing repetitions). Then${\rm det}(I+X) = \prod_{i=1}^{n}(1 + \lambda_{i}) \geq 1 + \prod_{i=1}^{n} \lambda_{i} = 1 + {\rm det}X.$Such an argument appears in some proofs by R. Brauer, though I do not know whether it originates with him. Later edit: Incidentally, I think that with the arithemetic-geometric mean inequality and a slightly more careful analysis, you can see by this approach that for$X$as above, you do have${\rm det}(I+X) \geq (1 +({\rm det}X)^{1/n})^{n}$(a special case of the inequality of Minkowski mentioned in the accepted answer). For set$d = {\rm det}X$. Let$s_{m}(\lambda_{1},\ldots ,\lambda_{n})$denote the$m$-th elementary symmetric function evaluated at the eigenvalues. Using the arithmetic-geometric mean inequality yields that$s_{m}(\lambda_{1},\ldots ,\lambda_{n}) \geq \left( \begin{array}{clcr} n\\m \end{array} \right)d^{m/n}$, so we obtain${\rm det}(I+X) \geq (1+d^{1/n})^{n}.\$

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