If A , B , and ( adj ( A ⁻¹ )+ adj ( B ⁻¹ ) ) are non-singular matrices of same order, then the inverse of A ( adj ( A ⁻¹ )+ adj ( B ⁻¹ ) )⁻¹ ~B , is equal to

Step-by-step solution

aligned & [ A ( adj ( A ⁻¹ )+ adj ( B ⁻¹ ) )⁻¹ ~B ]⁻¹ & ~B ⁻¹ ( adj ( ~A ⁻¹ )+ adj ( B ⁻¹ ) ) A ⁻¹ & ~B ⁻¹ adj ( ~A ⁻¹ ) A ⁻¹+ B ⁻¹ ( adj ( ~B ⁻¹ ) ) A ⁻¹ & ~B ⁻¹ | ~A ⁻¹ | I + | B ⁻¹ | IA ⁻¹ & ~B ⁻¹ | ~A | + A ⁻¹ | ~B | & adjB | B || A | + adj | A || B | & = 1 | ~A || B | ( adjB + adj A ) aligned