Let M be a 3×3 invertible matrix with real entries and let I denote the 3×3 identity matrix. If M−1=adj(adj M), then which of the following statements is/are ALWAYS TRUE?
A
M=I
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B
detM=1
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C
M2=I
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D
(adj M)2=I
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Solution
The correct option is D(adj M)2=I M−1=adj (adj (M))⋯(1)
and M−1=1|M|adj(M)⋯(2)
Now from (1) and (2) 1|M||M|2=|M|4⇒|M|=1
From (2) M−1=adj(M)⋯(3)
Using (1) M=adj(M)⋯(4)
From (3) and (4) I=(adj(M))2=(M)2