JEE Main 2022 · Mathematics · Matrices and Determinants · medium

The sum of all nine entries of A^10 is

Single correct option

Let

A=\begin{bmatrix}1&1&0\\0&1&1\\0&0&1\end{bmatrix}

. The sum of all nine entries of

A^{10}

Show answer and solution

Answer: (B) $68$

Write

A=I+N

with

N=\begin{bmatrix}0&1&0\\0&0&1\\0&0&0\end{bmatrix}

. Since

N^3=O

and

I

commutes with

N

, the binomial theorem gives

A^{10}=I+10N+\binom{10}{2}N^2=I+10N+45N^2

. Here

N^2

has a single non-zero entry

1

in position

(1,3)

. Adding entries:

I

contributes

3

10N

contributes

10+10=20

, and

45N^2

contributes

45

. The total is

3+20+45=68

. Omitting the

N^2

term wrongly gives

23

Original question written in the JEE Main 2022 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake