Let the third-order square matrix a satisfy (a + e) 3 = 0, find all eigenvalues of matrix A, where e is the third-order identity matrix

Let the third-order square matrix a satisfy (a + e) 3 = 0, find all eigenvalues of matrix A, where e is the third-order identity matrix

Let K be the eigenvalue of a and a be the eigenvector corresponding to K (a is not equal to zero vector). Then AA = Ka
Because (a + e) ^ 3 = 0
That is, a ^ 3 + 3A ^ 2 + 3A + e = 0
Right multiply a on both sides of the above formula to get:
k^3a+3k^2a+3ka+a=0
That is, (k ^ 3 + 3K ^ 2 + 3K + 1) a = 0
(k+1)^3a=0
Because a is not a zero vector, so (K + 1) ^ 3 = 0
So k = - 1 (triple eigenvector)