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Maths-

General

Easy

Question

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

If det A $\neq$ ±1, then A–1 exists and all its entries are non-integers
If det A = ±1, then A–1 exists and all its entries are integers
If det A = ±1, then A–1 need not exist
If det A = ±1, then A–1 exists but all its entries are not necessarily integers

hint Hint:

Since A has all integer elements, its cofactors will also be integers. So $A d j (A)$ will have integer elements
So if $∣ A ∣ = \pm 1$
$w e space h a v e space A to the power of negative 1 end exponent equals fraction numerator 1 over denominator open vertical bar A close vertical bar end fraction space space x space A d j left parenthesis A right parenthesis equals plus-or-minus 1 cross times A d j left parenthesis A right parenthesis A n d space h e n c e space A to the power of negative 1 end exponent space e x i s t s comma space a n d space h a s space a l l space t h e space e l e m e n t s space a s space i n t e g e r s.$ $T h e r e f o r e space A to the power of negative 1 end exponent space e x i s t s comma space a n d space h a s space a l l space t h e space e l e m e n t s space a s space i n t e g e r s.$

The correct answer is: If det A = ±1, then A–1 exists and all its entries are integers

GIven A be a square matrix all of whose entries are integers.We need to find which one of the following is true ?

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