# Determinant identities

In mathematics the determinant is an operator which has certain useful identities.

## Identities

${\displaystyle \det(I_{n})=1}$ where In is the n × n identity matrix.

${\displaystyle \det(A^{\rm {T}})=\det(A).}$

${\displaystyle \det(A^{-1})={\frac {1}{\det(A)}}=\det(A)^{-1},}$ provided ${\displaystyle A^{-1}}$ exists.

For square matrices A and B of equal size,

${\displaystyle \det(AB)=\det(A)\det(B).}$

${\displaystyle \det(cA)=c^{n}\det(A)}$ for an n × n matrix.

If A is a triangular matrix, i.e. ai,j = 0 whenever i > j or, alternatively, whenever i < j, then its determinant equals the product of the diagonal entries: ${\displaystyle \det(A)=a_{1,1}a_{2,2}\cdots a_{n,n}=\prod _{i=1}^{n}a_{i,i}.}$

### Schur complement

The following identity holds for a Schur complement of a square matrix:

The Schur complement arises as the result of performing a block Gaussian elimination by multiplying the matrix M from the right with the "block lower triangular" matrix

${\displaystyle L=\left[{\begin{matrix}I_{p}&0\\-D^{-1}C&I_{q}\end{matrix}}\right].}$

Here Ip denotes a p×p identity matrix. After multiplication with the matrix L the Schur complement appears in the upper p×p block. The product matrix is

{\displaystyle {\begin{aligned}ML&=\left[{\begin{matrix}A&B\\C&D\end{matrix}}\right]\left[{\begin{matrix}I_{p}&0\\-D^{-1}C&I_{q}\end{matrix}}\right]=\left[{\begin{matrix}A-BD^{-1}C&B\\0&D\end{matrix}}\right]\\&=\left[{\begin{matrix}I_{p}&BD^{-1}\\0&I_{q}\end{matrix}}\right]\left[{\begin{matrix}A-BD^{-1}C&0\\0&D\end{matrix}}\right].\end{aligned}}}

That is, we have effected a Gaussian decomposition

{\displaystyle {\begin{aligned}\left[{\begin{matrix}A&B\\C&D\end{matrix}}\right]&=\left[{\begin{matrix}I_{p}&BD^{-1}\\0&I_{q}\end{matrix}}\right]\left[{\begin{matrix}A-BD^{-1}C&0\\0&D\end{matrix}}\right]\left[{\begin{matrix}I_{p}&0\\D^{-1}C&I_{q}\end{matrix}}\right],\end{aligned}}}

The first and last matrices on the RHS have determinant unity, so we have

${\displaystyle {\rm {det}}\left|{\begin{matrix}A&B\\C&D\end{matrix}}\right|={\rm {det}}|D|\,{\rm {det}}|A-BD^{-1}C|.}$

This is Schur's determinant identity.