Bounds on determinants of perturbed diagonal matrices

We give upper and lower bounds on the determinant of a perturbation of the identity matrix or, more generally, a perturbation of a nonsingular diagonal matrix. The matrices considered are, in general, diagonally dominant. The lower bounds are best possible, and in several cases they are stronger than well-known bounds due to Ostrowski and other authors. If $A=I-E$ is an $n\times n$ matrix and the elements of $E$ are bounded in absolute value by $\epsilon \le 1\mathrm{/}n$, then a lower bound of Ostrowski (1938) is $\det (A)\ge 1-n\epsilon$. We show that if, in addition, the diagonal elements of $E$ are zero, then a best-possible lower bound is $$\det (A)\ge (1-(n-1)\epsilon )(1+\epsilon {)}^{n-1}\mathrm{.}$$ Corresponding upper bounds are respectively $$\det (A)\le (1+2\epsilon +n{\epsilon }^2{)}^{n\mathrm{/}2}$$ and $$\det (A)\le (1+(n-1){\epsilon }^2{)}^{n\mathrm{/}2}\mathrm{.}$$ The first upper bound is stronger than Ostrowski’s bound (for $\epsilon <1\mathrm{/}n$) $\det (A)\le (1-n\epsilon {)}^{-1}$. The second upper bound generalises Hadamard’s inequality, which is the case $\epsilon =1$. A necessary and sufficient condition for our upper bounds to be best possible for matrices of order $n$ and all positive $\epsilon$ is the existence of a skew-Hadamard matrix of order $n$.