Let A, B, C and D be n x n matrices, each with non-zero determinant. If ABCD =…
2004
Let A, B, C and D be n x n matrices, each with non-zero determinant. If ABCD = I, then B^-1 is
- A.
D^-1 C^-1 A^-1
- B.
CDA
- C.
ADC
- D.
Does not necessarily exist
Attempted by 18 students.
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Correct answer: B
Since A, B, C and D all have non-zero determinant, each matrix is invertible. Given ABCD = I, multiply on the left by A^-1 to get BCD = A^-1. Now multiply on the right by D^-1 C^-1 to isolate B: B = A^-1 D^-1 C^-1. Taking inverse on both sides reverses the order, so B^-1 = (A^-1 D^-1 C^-1)^-1 = C D A. Therefore, the correct answer is CDA.