Let A_k = bmatrix 1 & -1 & k 2 & 1 & 0 -1 & -1 & 1 bmatrix be a 3 3 matrix. If _ k=1 ^n ( adj A_k) = 655 , then the value of n is equal to

Step-by-step solution

First, we evaluate the determinant of the matrix A_k . (A_k) = 1(1 - 0) - (-1)(2 - 0) + k(-2 - (-1)) (A_k) = 1 + 2 - k = 3 - k We know that for a 3 3 matrix, ( adj A_k) = ( A_k)^2 . So, ( adj A_k) = (3 - k)^2 = (k - 3)^2 The given summation is: _ k=1 ^n (k - 3)^2 = 655 Let us write out the first few terms of this sum: For k = 1 , (1 - 3)^2 = 4 For k = 2 , (2 - 3)^2 = 1 For k = 3 , (3 - 3)^2 = 0 For k 4 , the terms are 1^2, 2^2, 3^2, , (n - 3)^2 . Thus, the sum can be written as: 4 + 1 + 0 + _ j=1 ^ n-3 j^2 = 655 5