Hadamard's Determinant Inequalities and Applications I

Problem

If $A=(a_{ij})\in M_n(\mathbb{R}),$ then

$\displaystyle (\det A)^2\le\prod_{j=1}^n\left(\sum_{i=1}^na_{ij}^2\right).$

(See, e.g., Hadamard's theorem on determinants.)

Problem 1, Solution

Consider a $3\times 3$ matrix $A=\left(\begin{array}{ccc}a&1&1\\1&b&1\\1&1&c\end{array}\right).$ One can verify that $\det A=2-a-b-c+abc.$ Thus, the required inequality is a direct consequence of Hadamard's First Theorem.

Problem 2, Solution

Consider a $4\times 4$ matrix $A=\left(\begin{array}{ccc}1&1&1&1\\1&a&0&0\\1&0&b&0\\1&0&0&c\end{array}\right).$ One can verify that $\det A=abc-a-b-c.$ Thus, the required inequality is a direct consequence of Hadamard's First Theorem.

Acknowledgment

The above is based on an article Application of Hadamard's Theorems to inequalities by Dan Sitaru and Leo Giugiuc that appeared in the Crux Mathematicorum (v 44, n 1, pp 25-27). I am very much indebted to Dan Sitaru for bringing this article to my attention.

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