Geodesic
A Determinantal Inequality for Positive Definite Matrices
Canadian mathematical bulletin, Tome 4 (1961) no. 1, pp. 57-62

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Let H = (Hi, j) (1 ≦ i, j ≦ n) be an nk × nk matrix with complex coefficients, where each Hi, j is itself a k × k matrix (n, k ≧ 2). Let |H| denote the determinant of H and let ∥H∥ = |(|H i, j|)| (1 ≦ i, j ≦ n ). The purpose of this note is to prove the following theorem.Theorem. If H is positive definite Hermitian then |H| ≦∥H∥. Moreover, |H| = ∥H∥ if and only if Hi, j = 0 whenever i ≠ j.The case n = 2 of this theorem is contained in [1]. 
Thompson, R. C. A Determinantal Inequality for Positive Definite Matrices. Canadian mathematical bulletin, Tome 4 (1961) no. 1, pp. 57-62. doi: 10.4153/CMB-1961-010-9
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     title = {A {Determinantal} {Inequality} for {Positive} {Definite} {Matrices}},
     journal = {Canadian mathematical bulletin},
     pages = {57--62},
     year = {1961},
     volume = {4},
     number = {1},
     doi = {10.4153/CMB-1961-010-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1961-010-9/}
}
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