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JEE Main · 2024 · 08 Apr 2024 · Shift 1 · Matrices

Let

A = [21 - 1 1]

. If the sum of the diagonal elements of

A^{13}

is

3^{n}

, then n is equal to ______.

Answer: 7

Solution

For

A = [21 - 1 1]

, the characteristic equation is

λ^{2} - 3 λ + 3 = 0

, so

A^{2} = 3 A - 3 I

and hence

A^{k} = 3 A^{k - 1} - 3 A^{k - 2}

. Taking traces,

t_{k} = 3 t_{k - 1} - 3 t_{k - 2}

with

t_{1} = 3, t_{2} = 3

. Iterating:

t_{3} = 0, t_{4} = - 9, t_{5} = - 27, t_{6} = - 54, t_{7} = - 81, t_{8} = - 81, t_{9} = 0, t_{10} = 243, t_{11} = 729, t_{12} = 1458, t_{13} = 2187 = 3^{7}

. So the trace of

A^{13}

is

3^{7}

, giving

n = 7

.

JEE Main · 2024 · 08 Apr 2024 · Shift 1 · Integer answer

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