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JEE Main · 2022 · 24 Jun 2022 · Shift 2 · Matrices

Let

S = {(- 1 0 a b); a, b \in {1, 2, 3, \dots 100}}

and let

T_{n} = {A \in S : A^{n (n + 1)} = I}

. Then the number of elements in

⋂_{n = 1}^{100} T_{n}

is ______.

Answer: 100

Solution

For

A = (- 1 0 a b)

, compute

A^{2} = (10 - a + ab b^{2})

. For

A^{n (n + 1)} = I

to hold for all

n

, we need

A^{2} = I

, which requires

b^{2} = 1

and

- a + ab = 0

. Since

b \in {1, \dots, 100}

and

b^{2} = 1

, we must have

b = 1

. Then

- a + a = 0

is satisfied for any

a \in {1, \dots, 100}

. Thus there are 100 such matrices.

JEE Main · 2022 · 24 Jun 2022 · Shift 2 · Integer answer

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