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 statement is/are ALWAYS TRUE ?
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a
M=I
b
det M=1
c
M2=I
d
(adjM)2=I
answer is B.
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Detailed Solution
det(M)≠0M−1=adj(adjM)M−1=det(M)⋅M since adjadj A=An-2AM−1M=det(M)⋅M2I=det(M)⋅M2 …(i)det(I)=(det(M))5 since kA=k3A if A is 3×3 matrix1=det(M) ⋯(ii) From (i) I=M2(adjM)2=adjM2=adjI=I