JEE Advanced 2018 Paper 1 · Mathematics · Matrices and Determinants · medium

Let A=beginpmatrix1&1&0\\0&1&1\\0&0&1endpmatrix and let I be the 3×3 identity matrix.

One or more options correct

Let

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

and let

I

be the

3\times3

identity matrix. Which of the following statements is (are) correct?

Show answer and solution

Answer: A, C

Write

A=I+N

with

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

, so

N^2

has a single entry

1

at position

(1,3)

and

N^3=0

. Since

I

and

N

commute,

A^n=I+nN+\binom n2N^2

. (A) The

(1,3)

entry of

A^{10}

\binom{10}{2}=45

, so it is true. (B)

A^{-1}=I-N+N^2

because

(I+N)(I-N+N^2)=I+N^3=I

, hence the

(1,3)

entry is

+1

, not

-1

, so it is false. (C)

A^4=I+4N+6N^2

has entry sum

3+4\cdot2+6\cdot1=17

, so it is true. (D)

(A-I)^2=N^2\neq0

, so the minimal polynomial is

(x-1)^3

and the statement is false.

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