  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
$\varepsilon \le 1/n$, then a lower bound of Ostrowski (1938) is $\det(A) \ge
1-n\varepsilon$. We show that if, in addition, the diagonal elements of $E$ are
zero, then a best-possible lower bound is \[\det(A) \ge
(1-(n-1)\varepsilon)\,(1+\varepsilon)^{n-1}.\] Corresponding upper bounds are
respectively \[\det(A) \le (1 + 2\varepsilon + n\varepsilon^2)^{n/2}\] and
\[\det(A) \le (1 + (n-1)\varepsilon^2)^{n/2}.\] The first upper bound is
stronger than Ostrowski's bound (for $\varepsilon < 1/n$) $\det(A) \le (1 -
n\varepsilon)^{-1}$. The second upper bound generalises Hadamard's inequality,
which is the case $\varepsilon = 1$.
  A necessary and sufficient condition for our upper bounds to be best possible
for matrices of order $n$ and all positive $\varepsilon$ is the existence of a
skew-Hadamard matrix of order $n$.
