We know that for an invertible matrix A
A−1=det(A)1.adJ(A)....(I)
For a 2×2 matrix A=[acbd]
A−1=det(A)1[d−c−ba]......(2)
We gave
A−1=det(A)1[d−c−ba] withA#0
Therefore from equition 1 we can conclude that
A−1=det(A)1.adJ(A)=det(A)1[d−c−ba].....(2)
Since we have
A−1=det(A)1[d−c−ba]
Therefore
A=[acbd]
A=[acbd]T....(4)
Also
adj(A)=[d−c−ba]=[d−b−ca]T....(5)
Hence, proved that (I)(ii) and (iii) are all true.
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