Q.

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

(adj⁡M)2=I

answer is B.

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Detailed Solution

det⁡(M)≠0M−1=adj⁡(adj⁡M)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(adj⁡M)2=adj⁡M2=adj⁡I=I
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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 ?