Question

Let M be a 3×3 matrix with real entries such that M2=M+2I, where I denotes that 3×3 identity matrix. If α,β and γ are eigenvalues of M such that αβγ=−4, then α+ B+γ is equal to

Answer 1

The sum of the eigenvalues α + β + γ of the matrix M, given that αβγ = -4 and M² = M + 2I, is equal to the trace of M, which is 1.

Step-by-step explanation:

The student is dealing with a linear algebra problem involving a 3×3 matrix M and its eigenvalues. Since M2 = M + 2I, and the product of eigenvalues αβγ is -4, we can use the properties of determinants and eigenvalues to find the sum of the eigenvalues, α + β + γ. The eigenvalues of a matrix are the roots of its characteristic polynomial, and for a 3×3 matrix, the sum of the eigenvalues is equal to the trace of the matrix (the sum of the diagonal entries). Moreover, the given equation suggests that the trace of M is 1 because the trace is an eigenvalue of M and it could be either +1, 0, or -1 to satisfy M2 = M + 2I. Hence, the sum of the eigenvalues, which is the trace of M, would be 1.