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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 ?

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 ? Correct Answer 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.

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