A) If\[\det (A)=\pm 1\], then\[{{A}^{-1}}\]exists but all its entries are not necessarily integers
B) If\[\det (A)\ne \pm 1\], then\[{{A}^{-1}}\].exists and all its entries are non-integers
C) If\[\det (A)=\pm 1\], then\[{{A}^{-1}}\]exists and all its entries are integers
D) If\[\det (A)=\pm 1\], then\[{{A}^{-1}}\]need not exist
Correct Answer: C
Solution :
As\[\det (A)=\pm 1,\,\,{{A}^{-1}}\]exists and \[{{A}^{-1}}=\frac{1}{\det (A)}(adj\,\,A)=\pm adj\,\,A\] All entries in \[adj\,(A)\] are integers. \[\therefore \]\[{{A}^{-1}}\]has integer entries.
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