Let A = bmatrix 1 & 1 & 1 & 1 0 & 1 & 1 & 1 0 & 0 & 1 & 1 0 & 0 & 0 & 1 bmatrix . If for some n N , the entry in the first row and fourth column of the matrix A^n is 364 , then the value of n is _______.

Step-by-step solution

Let A = I + N , where I is the 4 4 identity matrix and N = bmatrix 0 & 1 & 1 & 1 0 & 0 & 1 & 1 0 & 0 & 0 & 1 0 & 0 & 0 & 0 bmatrix . Calculating the powers of N , we get: N^2 = bmatrix 0 & 0 & 1 & 2 0 & 0 & 0 & 1 0 & 0 & 0 & 0 0 & 0 & 0 & 0 bmatrix N^3 = bmatrix 0 & 0 & 0 & 1 0 & 0 & 0 & 0 0 & 0 & 0 & 0 0 & 0 & 0 & 0 bmatrix and N^4 = O (the zero matrix). Using the binomial expansion since I and N commute: A^n = (I + N)^n = I + nN + n(n-1) 2 N^2 + n(n-1)(n-2) 6 N^3 The entry in the first row and fourth column of A^