Matrices & DeterminantsmediumFree

Matrices & Determinants: Let Pmatrix Pmatrix Let Pmatrix Pmatrix Equal

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

Question
Let M=(0ii0)M = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}, where i2=−1i^2 = -1, and let I=(1001)I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. Then I+M+M2+M3+⋯+M2010I + M + M^2 + M^3 + \cdots + M^{2010} is equal to:
A(0000)\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
B(0ii0)\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}correct
C(1ii1)\begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}
D(−100−1)\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}
Solution
Step 1: Compute powers of MM
M2=(−100−1)=−I,M3=−M,M4=IM^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+M2+M3=I+M−I−M=0I + M + M^2 + M^3 = I + M - I - M = 0
 Step 3: Account for all terms The sum has 2011 terms (n=0n = 0 to 20102010). Since 2011=4(502)+32011 = 4(502) + 3, the first 2008 terms cancel in complete groups of four, leaving:
M2008+M2009+M2010=I+M+M2=I+M−I=M=(0ii0)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
Similar questions
Matrices & Determinants · easy
If AA is a 3×33 \times 3 matrix and det⁡A=5\det A = 5, then det⁡(Adj(Adj A))\det\big(\text{Adj}(\text{Adj}\,A)\big) equals:
Matrices & Determinants · easy
The equation (12213434k)(xyz)=(000)\begin{pmatrix} 1 & 2 & 2 \\ 1 & 3 & 4 \\ 3 & 4 & k \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} has a solution (x,y,z)(x, y, z) besides (0,0,0)(0, 0, 0). The value of kk equals:
Matrices & Determinants · easy
AA is an involutory matrix given by A=(01−14−343−34)A = \begin{pmatrix}0 & 1 & -1\\ 4 & -3 & 4\\ 3 & -3 & 4\end{pmatrix}, then the inverse of A2\dfrac{A}{2} will be:
Matrices & Determinants · easy
A square matrix PP satisfies P2=I−PP^2 = I - P, where II is the identity matrix. If Pn=5I−8PP^n = 5I - 8P, then nn is:

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.

Sign up freeExplore doMath