Matrices & DeterminantsmediumFree

Matrices & Determinants — JEE Maths practice question

JEE Maths question with a full step-by-step solution.

Question

Let

M = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}

, where

i^2 = -1

, and let

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

. Then

I + M + M^2 + M^3 + \cdots + M^{2010}

is equal to:

\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}

\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}

correct

\begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}

\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}

Solution

Step 1: Compute powers of

M

M^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = -I, \quad M^3 = -M, \quad M^4 = I

The powers repeat with period 4. Step 2: Sum over one period

I + M + M^2 + M^3 = I + M - I - M = 0

Step 3: Account for all terms The sum has 2011 terms (

n = 0

2010

). Since

2011 = 4(502) + 3

, the first 2008 terms cancel in complete groups of four, leaving:

M^{2008} + M^{2009} + M^{2010} = I + M + M^2 = I + M - I = M = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}

Still stuck on this question?Ask your doubt on WhatsApp

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.