Let A and B are square matrices of order 2 such that A + a d j B T = 3 2 2 3 and A T - a d j B = - 2 - 1 - 1 - 2 , then A 2 + 2 A 3 + 3 A 4 + 5 A 5 is equal to (where M T and a d j ( M ) represent the transpose matrix and adjoint matrix of matrix M respectively and I represents the identity matrix of order 2 )

Step-by-step solution

Let, X = 3 2 2 3 ,   Y = - 2 - 1 - 1 - 2 A + a d j B T = X A T - a d j B = Y A - a d j B T = Y T 2 A = X + Y T = 1 1 1 1 ⇒ A = 1 2 1 1 1 1 2 a d j B T = X - Y T = 5 3 3 5 ⇒ a d j B T = 1 2 5 3 3 5 A 2 = 1 4 1 1 1 1 1 1 1 1 = 1 4 2 2 2 2 = 1 2 1 1 1 1 = A A 3 = A 4 = A 5 = A