Editing Determinant (section)

=== Determinant of an endomorphism ===
The above identities concerning the determinant of products and inverses of matrices imply that [[matrix similarity|similar matrices]] have the same determinant: two matrices ''A'' and ''B'' are similar, if there exists an invertible matrix ''X'' such that {{math|1=''A'' = ''X''<sup>−1</sup>''BX''}}. Indeed, repeatedly applying the above identities yields

:<math>\det(A) = \det(X)^{-1} \det(B)\det(X) = \det(B) \det(X)^{-1} \det(X) = \det(B).</math>

The determinant is therefore also called a [[similarity invariance|similarity invariant]]. The determinant of a [[linear transformation]]
:<math>T : V \to V</math>
for some finite-dimensional [[vector space]] ''V'' is defined to be the determinant of the matrix describing it, with respect to an arbitrary choice of [[basis (linear algebra)|basis]] in ''V''. By the similarity invariance, this determinant is independent of the choice of the basis for ''V'' and therefore only depends on the endomorphism ''T''.