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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 options is (are) always TRUE?

A
M=I
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B
det(M)=1
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C
M2=I
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D
(adj M)2=I
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Solution

The correct options are
B det(M)=1
C M2=I
D (adj M)2=I
Given, M1=adj(adj M)
|M1|=|adj(adj M)|
1|M|=|M|(n1)2 (|adj(adj A)=|A|(n1)2)

For n=3
1|M|=|M|4
|M|5=1
|M|=1 (1)

Now, M1=adj(adj M)
M1=|M|32M (adj(adj A)=|A|n2A)
M1=|M|M
M1=M [from equation (1)]
M×M1=M×M
I=M2
or, M2=I

Now, multiply adj(M) both sides
adj(M)MM=adj(M)I
Since Aadj(A)=|A|I, we have
(|M|I)M=adj(M)
IM=adj(M) (|M|=1)
M=adj(M)

Again, multiply adj(M)
Madj(M)=(adj(M))2
|M|I=(adj(M))2
I=(adj(M))2

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