Let A be a 3 3 matrix with real entries such that tr (A) = 2 , tr ( adj (A)) = -1 , and |A| = 3 , where adj (A) denotes the adjoint of matrix A . If B = A^4 - 2A^3 - A^2 - 2A + 5I , where I is the 3 3 identity matrix, then the value of tr (B) is equal to

Step-by-step solution

For a 3 3 matrix A , the characteristic equation is ^3 - tr (A) ^2 + S₂ - |A| = 0 , where S₂ is the sum of the principal minors of order 2 . A known property of the adjoint matrix is that its diagonal elements are exactly the cofactors C₁₁, C₂₂, C₃₃ of A . Thus, tr ( adj (A)) = C₁₁ + C₂₂ + C₃₃ = S₂ . We are given S₂ = -1 . Substituting the given values into the characteristic equation: ^3 - 2 ^2 - - 3 = 0 By the Cayley-Hamilton theorem, A satisfies this equation: A^3 - 2A^2 - A - 3I = 0 We are given the matrix poly