Q

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

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

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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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