\[ \DeclareMathOperator{\sgn}{sgn} \]

Let \(A\) be a \(n\times n\)-matrix. Then the determinant is defined by

\begin{equation*} \det A = \sum_{\sigma \in S_n} \sgn(\sigma) a_{1,\sigma(1)}\cdots a_{1,\sigma(n)}, \end{equation*}

where \(S_n\) is the group of permutations of the set \(\{1,\ldots,n\}\) and \(\sgn \sigma\) the parity of \(\sigma\) .

Properties Link to heading

• \(\det A^T = \det A\)
• \(\det(AB)=\det A \det B\)
• \(\det A^{-1}= (\det A)^{-1}\)

Inequalities for positive semidefinite matrices Link to heading

Let \(A\), \(B\) positive semidefinite . Then

• \(\det A \ge 0\)
• \(A\le B \implies \det A \le \det B\)

See also Link to heading

• determinant of a linear map