Prove that determinant of a matrix A is the product of its eigenvalues

To prove that the determinant of a matrix A is the product of its eigenvalues, we can follow these steps:

Step 1: Start with a square matrix A of size n x n.

Step 2: Find the eigenvalues of matrix A. To do this, solve the characteristic equation det(A - λI) = 0, where λ is the eigenvalue and I is the identity matrix of size n x n.

Step 3: Once you have found all the eigenvalues, denote them as λ1, λ2, ..., λn.

Step 4: Calculate the product of the eigenvalues, which is given by λ1 * λ2 * ... * λn.

Step 5: Compute the determinant of matrix A, denoted as det(A).

Step 6: Show that the product of the eigenvalues, λ1 * λ2 * ... * λn, is equal to the determinant of matrix A, det(A).

By following these steps, you can prove that the determinant of a matrix A is indeed the product of its eigenvalues.