Question

Show that if A and B are similar nxn matrices, then det(A)=det(B).

Answer 1

Explanation:

To prove it we just use the definition of similar matrices and properties of determinants:

If
`A,B` are similar matrices, then there is an invertible matrix
`C`, such that
`A=C^(-1)BC}` (that's the definition of matrices being similar). And so we compute the determinant of such matrix to get:

`det(A)=det(C^(-1)BC)=det(C^(-1))det(B)det(C)`

`=(1)/(det(C))det(B)det(C)=det(B)`

(Determinant of a product of matrices is the product of their determinants, and the determinant of
`C^(-1)` is just
`(1)/(det(C))`)