Let A = pmatrix -2 & 1 & 0 0 & -1 & 1 1 & 0 & 0 pmatrix . If A^3 + A^2 + A + I = O , where I is the 3 3 identity matrix and O is the null matrix, then the value of + - is equal to

Step-by-step solution

The characteristic equation of a 3 3 matrix A is given by: ^3 - ( tr (A)) ^2 + (M₁₁ + M₂₂ + M₃₃) - (A) = 0 where M₁₁, M₂₂, M₃₃ are the principal minors of A . For the given matrix A = pmatrix -2 & 1 & 0 0 & -1 & 1 1 & 0 & 0 pmatrix : 1) Trace of A : tr (A) = -2 + (-1) + 0 = -3 2) Sum of principal minors: M₁₁ = vmatrix -1 & 1 0 & 0 vmatrix = 0 M₂₂ = vmatrix -2 & 0 1 & 0 vmatrix = 0 M₃₃ = vmatrix -2 & 1 0 & -1 vmatrix = 2 - 0 = 2 Sum = 0 + 0 + 2 = 2 3) Determinant of A : (A) = -2(0 - 0) - 1(0 - 1) + 0 = 1 Substitutin