If the eigenvalues of a matrix are 325 find the characterist

If the eigenvalues of a matrix are 3,-2,5, find the characteristic polynomial in two ways. How does this extend to the matrices that are 4x4

Solution

We are only advised that the eigenvalues of a matrix are 3,-2,5. The size of the matrix is not advised. If these are the eigenvalues of a 3x3 matrix, then its characteristic polynomial is unique and is (-3)(+2)(-5) = 0. However, if these are eigenvalues of a 4x4 matrix, then one of these eigenvalues has multiplicity 2. Then the characteristic polynomial can be (-3)²(+2)(-5) = 0 or, (-3)(+2)²(-5) = 0 or, (-3)(+2)(-5)² = 0 depending upon which eigenvalue, 3,-2 or 5 has multiplicity 2