Question

Let A, B, C, D be 3 x 3 matrices each with non-zero determinant and ABCD = I, where I is the identity matrix then C^-1 is equal to ?
1. DAB
2. AD
3. BD
4. CD

Answer 1

Correct Answer - Option 1 : DAB

Concept:

Properties of inverse

• P^-1I = P^-1
• PP^-1 = P^-1P = I
• (PQR)^-1 = R^-1Q-1P-1     
• (P^-1)^-1 = P

Calculation:

Given that, ABCD = I

⇒ A^-1ABCD = A^-1I = A^-1      (∴ A-1I = A-1)

⇒ BCD = A^-1                       (∴ A-1A = I)  

⇒ B^-1BCD = B^-1A^-1               (multiply both side by B^-1)

⇒ CD =  B-1A-1                    (∴ B-1B = I )

⇒ CDD^-1 = B-1A-1D^-1          (multiply both side by D-1)

⇒ C = B-1A-1D-1                  (∴ D-1D = I )

⇒ C^-1 = (B-1A-1D-1 )^-1

⇒  C-1 = (D-1)^-1(A-1)^-1(B-1)-1   (∴ PQR)-1 = R-1Q-1P-1)

⇒ C-1 = DAB

Hence, option 1 is correct.