Let A, B and A + B are non-singular matrices of order 3 x 3 satisfying A^-1 + B^-1 = (A + B)^-1 and |AB^-1| is R then value of |A|/|B| is
Step-by-Step Solution
Key Concept: Given A^-1 + B^-1 = (A + B)^-1. Multiply by A on the left and B on the right: A(A^-1 + B^-1)B = A(A + B)^-1 B. This simplifies to B + A = A(A + B)^-1 B. Let X = A + B. Then X = AX^-1 B, so X^2 = AB. Taking determinants, |A+B|^2 = |A||B|. Since A^-1 + B^-1 = (A+B)^-1, we have (A+B)(A^-1 + B^-1) = I, which implies I + AB^-1 + BA^-1 + I = I, so AB^-1 + BA^-1 = -I. This leads to the conclusion that |A|/|B| = 1.
Given A^-1 + B^-1 = (A + B)^-1. Multiplying by A on the left and B on the right: A(A^-1 + B^-1)B = A(A + B)^-1 B. B + A = A(A + B)^-1 B. Let A + B = X. Then X = AX^-1 B => X^2 = AB. Taking determinant on both sides, |A+B|^2 = |A||B|. Also, A^-1 + B^-1 = (A+B)^-1 => (A+B)(A^-1 + B^-1) = I => I + AB^-1 + BA^-1 + I = I => AB^-1 + BA^-1 = -I. This implies |A|/|B| = 1.
Correct Answer: 1