Question
Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. M − 1 = a d j a d j   M , then which of the following statements is/are ALWAYS TRUE?
JEE Advanced
Mathematics
Matrices
2020
JEE Advanced 2020 (Paper 1)
JEE Advanced Mathematics Question (2020) — Solution
Options
- A. M = I
- B. det M = 1
- C. M 2 = I
- D. a d j   M 2 = I
Answer
D. a d j   M 2 = I
Step-by-step solution
Given M − 1 = adj adjM ∴ adj M ⋅ M − 1 = adj M  . adj  adjM ⇒adj M . M − 1 = AdjM I ⇒adj M = M 2 M   . . . . . . . . . 1 ⇒ a d j   M = M 2 M = M 6 M ⇒ M 2 = M 7 ⇒ M ≠ 0 ,   M = 1   . . . . . . . . . . . . . 2 By equations 1 &   2 , we get adj M = M   . . . . . . . . . . . 3 ⇒M . adjM = M 2 ⇒ M I = M 2 ⇒ M 2 = I Now, by equation 3 , adj M = M ⇒ adj  M 2 = M 2 = I
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