JEE Advanced 2019 Paper 1 · Mathematics · Matrices and Determinants · easy

For a positive integer n, the (1,3) entry of M^n+M^-n equals 1444. Then n equals

Single correct option

Let

M=\begin{pmatrix}1&2&0\\0&1&2\\0&0&1\end{pmatrix}

. For a positive integer

n

, the

(1,3)

entry of

M^{n}+M^{-n}

equals

1444

. Then

n

equals

Show answer and solution

Answer: (C) $19$

Write

M=I+N

with

N=\begin{pmatrix}0&2&0\\0&0&2\\0&0&0\end{pmatrix}

, so

N^2

has a single non-zero entry

4

at position

(1,3)

and

N^3=0

. The binomial series terminates:

M^{n}=I+nN+\binom n2N^2

and

M^{-n}=I-nN+\binom{-n}{2}N^2=I-nN+\frac{n(n+1)}2N^2

. Hence the

(1,3)

entries are

4\binom n2=2n(n-1)

and

2n(n+1)

. Their sum is

4n^2

. Setting

4n^2=1444

gives

n^2=361

, so

n=19

. The other options give

4n^2=1156

1296

and

1600

, none of which equals

1444

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