# Hadamard's Determinant Inequalities and Applications I

### Problem

### Hadamard's First Theorem

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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