Question

which of the following statements is/are TRUE?

Answer 1

Answer:

Maybe (c) option.

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Answer 2

Let M be a 3 × 3 invertible matrix with real entries and I debtors the 3 × 3 identity matrix. if M¯¹ = adj(adj M), then which of the following statements is/are always TRUE ?

solution : M be a 3 × 3 invertible matrix

means, det (M) ≠ 0

and given, M¯¹ = adj(adj M)

adj M. M¯¹ = adj M. adj (adj M)

⇒adj M. M¯¹ = |adj M| I [ as we know, A¯¹ = adj(A)/|A| ]

⇒adj M M¯¹ = |M|³¯¹ = |M|²

⇒adj M = |M|²M........(1)

⇒|adj M| = ||M|²M| = |M|⁶ |M| = |M|^7

⇒|M|² = |M|^7

⇒|M| = 1 ......(1)

from equation (1) we get, adj M = (1)² M

adj M = M

M. adj M = M.M = M²

⇒M. M¯¹ |M| = M²

⇒|M| I = M²

⇒1 × I = M² [ from equation (2) ]

⇒M² = I

again from equations (1) and (2) we get,

(adj M)² = M² = I

Therefore the correct options are option (B) , (C) and (D)