Question

Let A and B be two similar n x n matrices. Prove the following properties: (a) det(A) = det(B) (b) The characteristic polynomials of A and B are the same.

Answer 1

The determinant of similar matrices is equal, and the characteristic polynomials of similar matrices are the same.

Step-by-step explanation:

(a) Proving that det(A) = det(B):

Since A and B are similar n x n matrices, they have the same number of rows and columns as well as the same ratio of corresponding elements. If we expand the determinant of A using cofactor expansion along the first row, and the determinant of B using cofactor expansion along the first row, we can see that each term in the expansion of det(A) corresponds to the same term in the expansion of det(B). Therefore, the determinants of A and B are equal: det(A) = det(B).

(b) Proving that the characteristic polynomials of A and B are the same:

The characteristic polynomial of a matrix A is defined as det(A - λI), where λ is an eigenvalue of A and I is the identity matrix. Since A and B are similar matrices, they have the same eigenvalues. Therefore, the characteristic polynomials of A and B are the same.