The GATE Grind

GATE 2016 EC – Question 11

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Let $M^4 = I$, (where $I$ denotes the identity matrix) and $M \neq I$, $M^2 \neq I$ and $M^3 \neq I$. Then, for any natural number $k$, $M^{-1}$ equals:

  1. $M^{4k+1}$
  2. $M^{4k+2}$
  3. $M^{4k+3}$
  4. $M^{4k}$

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Show answer and explanation

Correct answer: (C) $M^{4k+3}$

Explanation

From $M^4 = I$ we get $M \cdot M^3 = I$, so $M^{-1} = M^3$. Multiplying by $(M^4)^k = I$ does not change it, so $M^{-1} = M^3 (M^4)^k = M^{4k+3}$.