Let A, B and C be three 2 2 matrices with real entries such that B=(I+A)⁻¹ and A + C = I . If BC = [ array cc 1 & -5 -1 & 2 array ] and CB [ array l x₁ x₂ array ]= [ array c 12 -6 array ] , then x₁+x₂ is

Step-by-step solution

From B = (I + A)⁻¹ we get B(I + A) = I , so BA = I - B . From A + C = I : C = I - A . Calculating CB = C(I+A)⁻¹ = (I-A)(I+A)⁻¹ . Since B(I+A) = I , we have (I+A) = B⁻¹ , so C = (I-A)B⁻¹ . Multiplying by B : CB = (I-A) . However, using the given BC = bmatrix 1 & -5 -1 & 2 bmatrix and B(I+A) = I implies B - BA = B - (I-B) = 2B - I = BC , i.e. BC = 2B - I . Therefore B = 1 2 (BC + I) = 1 2 bmatrix 2 & -5 -1 & 3 bmatrix = bmatrix 1 & -5/2 -1/2 & 3/2 bmatrix . Also CB = 2B - I = bmatrix 1 & -5 -1 & 2 bmatrix . From CB b