Editing Determinant (section)

=== Sylvester's determinant theorem ===
[[Sylvester's determinant theorem]] states that for ''A'', an {{math|''m'' × ''n''}} matrix, and ''B'', an {{math|''n'' × ''m''}} matrix (so that ''A'' and ''B'' have dimensions allowing them to be multiplied in either order forming a square matrix):

:<math>\det\left(I_\mathit{m} + AB\right) = \det\left(I_\mathit{n} + BA\right),</math>

where ''I''<sub>''m''</sub> and ''I''<sub>''n''</sub> are the {{math|''m'' × ''m''}} and {{math|''n'' × ''n''}} identity matrices, respectively.

From this general result several consequences follow.
{{ordered list
| list-style-type=lower-alpha
| For the case of column vector ''c'' and row vector ''r'', each with ''m'' components, the formula allows quick calculation of the determinant of a matrix that differs from the identity matrix by a matrix of rank 1:
:<math>\det\left(I_\mathit{m} + cr\right) = 1 + rc.</math>
| More generally,<ref>Proofs can be found in http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/proof003.html</ref> for any invertible {{math|''m'' × ''m''}} matrix ''X'',
:<math>\det(X + AB) = \det(X) \det\left(I_\mathit{n} + BX^{-1}A\right),</math>
| For a column and row vector as above:
: <math>\det(X + cr) = \det(X) \det\left(1 + rX^{-1}c\right) = \det(X) + r\,\operatorname{adj}(X)\,c.</math>
| For square matrices <math>A</math> and <math>B</math> of the same size, the matrices <math>AB</math> and <math>BA</math> have the same characteristic polynomials (hence the same eigenvalues).
}}

A generalization is <math>\det\left(Z + AWB\right) = \det\left( Z\right) \det\left(W \right) \det\left(W^{-1} + B Z^{-1} A\right)</math>(see  [[Matrix determinant lemma]]), where ''Z'' is an {{math|''m'' × ''m''}} invertible matrix and ''W'' is an {{math|''n'' × ''n''}} invertible matrix.