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Question

Let M be a 3×3 invertible matrix with real entries and let I denote the 3×3 identity matrix. If M1=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
M1=adj (adj (M))(1)
and M1=1|M|adj(M)(2)
Now from (1) and (2)
1|M||M|2=|M|4|M|=1
From (2)
M1=adj(M)(3)
Using (1)
M=adj(M)(4)
From (3) and (4)
I=(adj(M))2=(M)2

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