Question

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

Answer 1

Step-by-step 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:

$det(A)=det(C^{-1}BC)=det(C^{-1})det(B)det(C)$

$=\frac{1}{det(C)}det(B)det(C)=det(B)$

(Determinant of a product of matrices is the product of their determinants, and the determinant of $C^{-1}$ is just $\frac{1}{det(C)}$)