Step-by-step explanation:
To prove it we just use the definition of similar matrices and properties of determinants:
If [tex] A,B[/tex] are similar matrices, then there is an invertible matrix [tex]C[/tex], such that [tex] A=C^{-1}BC}[/tex] (that's the definition of matrices being similar). And so we compute the determinant of such matrix to get:
[tex]det(A)=det(C^{-1}BC)=det(C^{-1})det(B)det(C)[/tex]
[tex]=\frac{1}{det(C)}det(B)det(C)=det(B)[/tex]
(Determinant of a product of matrices is the product of their determinants, and the determinant of [tex]C^{-1}[/tex] is just [tex]\frac{1}{det(C)}[/tex])
