Let A = bmatrix 1 & 0 & -1 3 & 4 & 5 0 & -6 & -7 bmatrix . If A⁻¹ = 1 (A^2 + A + I) , where I is the identity matrix of order 3 3 and , , are integers, 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 + S₂ - |A| = 0 , where S₂ is the sum of the principal minors of order 2 . First, find the trace of A : tr (A) = 1 + 4 + (-7) = -2 Next, find S₂ = C₁₁ + C₂₂ + C₃₃ : C₁₁ = (4)(-7) - (5)(-6) = -28 + 30 = 2 C₂₂ = (1)(-7) - (-1)(0) = -7 C₃₃ = (1)(4) - (0)(3) = 4 S₂ = 2 - 7 + 4 = -1 Now, find the determinant of A : |A| = 1(-28 + 30) - 0 + (-1)(-18 - 0) = 2 + 18 = 20 The characteristic equation is: ^3 - (-2) ^2 + (-1) - 20 = 0 ^3 + 2 ^2 - - 20 = 0 By