JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Matrices

Let

A = ∣ a_{ij} ∣

be a real matrix of order

3 \times 3

, such that

a_{i 1} + a_{i 2} + a_{i 3} = 1

, for

i = 1, 2, 3

. Then, the sum of all the entries of the matrix

A^{3}

is equal to:

Answer: (B)

Solution

Given that the sum of entries in each row of

A

is 1, we can write

A 111 = 111

.

This means

111

is an eigenvector of

A

with eigenvalue 1.

Therefore,

A^{3} 111 = 1^{3} 111 = 111

.

This means the sum of entries in each row of

A^{3}

is also 1. Since

A^{3}

is a

3 \times 3

matrix with 3 rows, the sum of all entries is

3 \times 1 = 3

JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Single correct option