Let A = bmatrix 3 & -2 & -2 1 & 0 & -1 2 & -2 & -1 bmatrix and C = A - 2I , where I is the identity matrix of order 3 . If C^6 = A + I for some scalars and , then the value of - is equal to _____ .

Step-by-step solution

Given A = bmatrix 3 & -2 & -2 1 & 0 & -1 2 & -2 & -1 bmatrix . First, we compute A^2 : A^2 = bmatrix 3 & -2 & -2 1 & 0 & -1 2 & -2 & -1 bmatrix bmatrix 3 & -2 & -2 1 & 0 & -1 2 & -2 & -1 bmatrix = bmatrix 3 & -2 & -2 1 & 0 & -1 2 & -2 & -1 bmatrix = A Thus, A is an idempotent matrix, meaning A^k = A for all integers k 1 . We are given C = A - 2I . Since A and I commute, we can use the binomial theorem to expand C^6 : C^6 = (A - 2I)^6 = _ k=0 ⁶ ⁶C_ k A^k (-2I)^ 6-k Separating the k=0 term and using A^k = A for k 1 :