Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

  • Option 1)

    If det\; A=\pm 1,then\; A^{-1} need not exist

  • Option 2)

    If det\; A=\pm 1,then\; A^{-1}  exists  but all its entries are not necessarily integers

  • Option 3)

    If det\; A\neq \pm 1,then\; A^{-1} exists and all its entries are non­integers

  • Option 4)

    If det\; A=\pm 1,then\; A^{-1} exists and all its entries are integers

 

Answers (1)

As we learnt in 

Inverse of a matrix -

 A^{-1}=\frac{1}{\left | A \right |}\cdot adjA

-

 

 A^{-1}= \frac{adj\left ( A \right )}{|A|}

So\, \, \, \, for\ |A|\neq \pm 1

A^{-1} exist and all its entries are non integers.


Option 1)

If det\; A=\pm 1,then\; A^{-1} need not exist

Incorrect Option

 

Option 2)

If det\; A=\pm 1,then\; A^{-1}  exists  but all its entries are not necessarily integers

Incorrect Option

 

Option 3)

If det\; A\neq \pm 1,then\; A^{-1} exists and all its entries are non­integers

Incorrect Option

 

Option 4)

If det\; A=\pm 1,then\; A^{-1} exists and all its entries are integers

Correct Option

 

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