Let A = [ array cc 1 & 3 2 1 & 2 array ], B = [ array cc 4 & -3 -2 & 2 array ] and C _ r = [ array cc r 3^ r & 2^ r 0 & ( r -1) 3^ r array ] be given matrices. If _ r =1 ⁵⁰ tr . (( AB )^ r C _ r )=3+ a 3^ b where tr .( A ) denotes trace of matrix A , then find the value of ( a + b ) . [Where a and b are relatively prime.]

Step-by-step solution

We have A B= [ array cc 1 & 3 2 1 & 2 array ] [ array cc 4 & -3 -2 & 2 array ]= [ array ll 1 & 0 0 & 1 array ]= I aligned & (A B)^1 C₁=C₁,(A B)^2 C₂=C₂ and so on. & tr (C_r )=r 3^r+(r-1) 3^r=(2 r-1) 3^r aligned _ r =1 ⁵⁰ tr (( AB )^ r C _ r )= tr (( AB )^1 C ₁ )+ tr (( AB )^2 C ₂ )+ . .+ tr (( AB )⁵⁰ C ₅₀ )= S (Let) aligned & S= tr (C₁ )+ tr (C₂ )+ .+ tr (C₅₀ ) & S=1 3^1+3 3^2+5 3^3+ +99 3⁵⁰ & 3 S= 1 3^2+3 3^3+ +97 3⁵⁰+99 3⁵¹ aligned aligned - & 2 S=1 3+2 3^2+2 3^3+ +2 3⁵⁰-99 3⁵¹ & =-3+2 3+2 3^2+ +2 3⁵⁰-99 3⁵¹ & =-