JEE Main2008MathematicsMatricesActual
Let A be a square matrix all of whose entries are integers. Then which one of the following is true?
Options
- AIf det A= 1 , then A⁻¹ exists but all its entries are not necessarily integers
- BIf det A 1 , then A⁻¹ exists and all its entries are non-integers
- CIf det A= 1 , then A⁻¹ exists and all its entries are integers
- DIf det A= 1 , then A⁻¹ need not exist
Correct answer
C. If det A= 1 , then A⁻¹ exists and all its entries are integers
Step-by-step solution
Each entry of A is integer, so the cofactor of every entry is an integer and hence each entry in the adjoint of matrix A is integer. Now det A= 1 and A⁻¹= 1 det (A) ( adj A) all entries in A ⁻¹ are integers.