If the matrix A= ( array cc 1 & -1 -1 & 1 array ) then A^ n+1 =

Step-by-step solution

Given A = pmatrix 1 & -1 -1 & 1 pmatrix . Calculate A^2 : A^2 = pmatrix 1 & -1 -1 & 1 pmatrix pmatrix 1 & -1 -1 & 1 pmatrix = pmatrix 1+1 & -1-1 -1-1 & 1+1 pmatrix = pmatrix 2 & -2 -2 & 2 pmatrix = 2 pmatrix 1 & -1 -1 & 1 pmatrix = 2A . Calculate A^3 : A^3 = A^2 A = (2A) A = 2A^2 = 2(2A) = 2^2 A . By induction, assume A^k = 2^ k-1 A . Then A^ k+1 = A^k A = (2^ k-1 A) A = 2^ k-1 A^2 = 2^ k-1 (2A) = 2^k A . Thus, A^n = 2^ n-1 A for n 1 . Substituting n+1 for n , we get A^ n+1 = 2^ (n+1)-1 A = 2^n A . Therefore, A^ n+