Eigenvalues

[k10sstar]

Let A be an n x n matrix. Show that det(A) is the product of all the roots of the characteristic polynomial of A.

[lurflurf]

State you definitions of det(A) and the characteristic polynomial of A, and show what you have done so far. If your definition of det(A) is the product of all the roots of the characteristic polynomial of A, you are done.

[HallsofIvy]

The quick proof is that any matrix is similar to either a diagonal matrix or a Jordan matrix with its eigenvalues on the main diagonal. In either case, the determinant is just the product of the numbers on the main diagonal- the product of the eigenvalues.

The difficulty with you not showing any attempt yourself is that we don't know what definitions and theorems you have available to prove this.