Question

Why we are putting determinant equal to zero to find the eigenvalue?

Answer 1

We find the eigenvalues by solving Ax = lambda x for the vector lambda that makes that equation true (that’s the definition of an eigenvalue). That’s the same as solving Ax — lambda x = 0, or (A — lambda)x = 0. This implies the matrix (A — lambda) is singular or not invertible. So the determinant of (A — lambda) must be zero. That means we can find the eigenvalues by setting the determinant of (A — lambda) equal to zero and solving for lambda (this is called the “characteristic polynomial”).