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JEE Main 2020
Matrices & Determinants
Matrices and Determinants
Medium

Question

Let M = \left[ {\matrix{ 0 & { - \alpha } \cr \alpha & 0 \cr } } \right], where α\alpha is a non-zero real number an N=∑k=149M2kN = \sum\limits_{k = 1}^{49} {{M^{2k}}} . If (I−M2)N=−2I(I - {M^2})N = - 2I, then the positive integral value of α\alpha is ____________.

Answer: 0

Solution

Key Concepts Used

This problem elegantly combines concepts from Matrix Algebra and Geometric Progressions (GP). A strong understanding of these areas is crucial for solving it efficiently. The core ideas we will leverage are:

  1. Matrix Multiplication and Powers: How to multiply matrices and raise a matrix to an integer power.
  2. Identity Matrix (II): A special square matrix that acts like the number '1' in scalar multiplication, meaning A⋅I=I⋅A=AA \cdot I = I \cdot A = A for any matrix AA (of compatible dimensions). A key property is Ik=II^k = I for any positive integer kk.
  3. Scalar Multiple of a Matrix: When a matrix is multiplied by a real number (
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