Let A = bmatrix & & bmatrix be a 2 2 real matrix such that + =1 and - =0 . If (I+A)^5 + (I-A)^5 = bmatrix 92 & 60 -90 & -58 bmatrix , where I is the 2 2 identity matrix, then the value of ^2 + ^2 + ^2 + ^2 is:

Step-by-step solution

We are given that tr (A) = + = 1 and (A) = - = 0 . By the Cayley-Hamilton theorem, every square matrix satisfies its own characteristic equation: A^2 - tr (A)A + (A)I = 0 A^2 - A = 0 A^2 = A . Since A^2 = A , it follows that A^k = A for all integers k 1 . Now, expand (I+A)^5 using the binomial theorem: (I+A)^5 = I + ⁵C₁A + ⁵C₂A^2 + ⁵C₃A^3 + ⁵C₄A^4 + ⁵C₅A^5 Since A^k = A , this simplifies to: (I+A)^5 = I + (5 + 10 + 10 + 5 + 1)A = I + 31A . Similarly, expand (I-A)^5 : (I-A)^5 = I - ⁵C₁A + ⁵C₂A^2 - ⁵C₃A^3 + ⁵C₄A^4 -