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NDA 1 2025 Maths, question 25

Topic: Matrices and determinants

Q25

Consider the following in respect of a non-singular matrix MM:

I. ∣M2∣=∣M∣2\left| M^{2} \right| = |M|^{2}

II. ∣M∣=∣M−1∣|M| = \left| M^{- 1} \right|

III. ∣M∣=∣MT∣|M| = \left| M^{T} \right|

How many of the above are correct?

  1. (a) None
  2. (b) One
  3. (c) TwoCorrect answer
  4. (d) All three

Solution.

Answer: (c) Two

  1. I follows from determinant multiplication: ∣M2∣=∣M∣∣M∣\left| M^{2} \right| = |M||M|.
  2. III holds because transposition does not change a determinant.
  3. For II, the actual identity is ∣M−1∣=1/∣M∣\left| M^{- 1} \right| = 1/|M|, which need not equal ∣M∣|M|.
  4. Thus only I and III are universally correct.