Determinant ($\det$)

The determinant is the alternating multilinear volume scale of a square matrix; it is nonzero exactly for invertible matrices.

The block matrix $M=\left(\begin{array}{cc}A& -B\\ B& A\end{array}\right)$ is similar over $\mathbb{C}$ to $\mathrm{diag}(A+iB,A-iB)$ via $T=\left(\begin{array}{cc}I& I\\ -iI& iI\end{array}\right)$. Therefore its determinant is $\det (A+iB)\det (A-iB)$ for arbitrary square matrices $A,B$. The argument uses a constant change-of-basis matrix and multiplicativity of the determinant.

For a differentiable invertible matrix $A(t)$, $\frac{d}{dt}\log \det A=\mathrm{tr}(A^{-1}\dot{A})$ wherever a consistent logarithm is defined. In particular a positive spatial metric tensor satisfies $h^{ij}{\dot{h}}_{ij}=\dot{h}/h$. This separates its local volume change from shape change.

For an invertible matrix $A$ and any matrix $H$,Indeed $\det (A+tH)=\det (A)\det (I+t{A}^{-1}H)$, and multilinearity shows that the linear coefficient of $\det (I+tB)$ is $\mathrm{tr}B$. At a singular matrix the derivative is instead written $\mathrm{tr}(\mathrm{adj}(A)H)$; this polynomial formula holds for all $A$.

On positive-definite matrices, the log-determinant has differential $d\log \det A=\mathrm{Tr}(A^{-1}dA)$ and is a strictly concave function.

The determinant is linear in each row and in each column when all the others are held fixed.

The determinant is multiplicative: for square matrices of the same size, $\det (AB)=\det (A)\det (B)$.

For an $n$ by $n$ matrix,

For diagonal entries $a_i$, superdiagonal entries $b_i$, and subdiagonal entries $c_i$, the leading principal determinants satisfy