KVPY 2020-21 SB SX Exam Previous Papers

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Question : 8 of 120

Marks: +1, -0

Let

A

be a

2 \×2

matrix of the form

A=[\table a , b ;1 , 1]

,where

a, b

are integers and

-50 \≤ b \≤ 50

. The number of such matrices

A

such that

A^{-1}

, the inverse of

A

, exists and

A^{-1}

contains only integer entries is

101
200
202
$101^{2}$

Solution:

\| {A}| \≠0 \⇒{a}-{b} \≠0

\⇒{a} \≠{b}

......(i)
Also,

\ A^{-1}={1}{a-b}[\table 1 , -1 ; -b ,a ]^{{T}}

= {1}/{a-b}[\table 1 , -b ; -1 ,a ]

Thus,

{a}-{b}=1

or

-1

......(ii)
So, required number of pairs

({a}, {b})

is

101 \×2=202

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