Let A = bmatrix 1 & 2 & 0 & 1 & 2 0 & 0 & 1 bmatrix , where is a real number. If the sum of all the elements of the matrix A¹⁰ is 343 , then the value of is equal to

Step-by-step solution

We can write the matrix A as A = I + N , where I is the identity matrix and N = bmatrix 0 & 2 & 0 & 0 & 2 0 & 0 & 0 bmatrix . Now, we compute the powers of N : N^2 = bmatrix 0 & 2 & 0 & 0 & 2 0 & 0 & 0 bmatrix bmatrix 0 & 2 & 0 & 0 & 2 0 & 0 & 0 bmatrix = bmatrix 0 & 0 & 4 0 & 0 & 0 0 & 0 & 0 bmatrix N^3 = bmatrix 0 & 0 & 4 0 & 0 & 0 0 & 0 & 0 bmatrix bmatrix 0 & 2 & 0 & 0 & 2 0 & 0 & 0 bmatrix = bmatrix 0 & 0 & 0 0 & 0 & 0 0 & 0 & 0 bmatrix Since N^3 = 0 , N is a nilpotent matrix of order 3. Using the binomial exp