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

I. $∣ M^{2} ∣ = ∣ M ∣^{2}$

II. $∣ M ∣ = ∣ M^{- 1} ∣$

III. $∣ M ∣ = ∣ M^{T} ∣$

How many of the above are correct?

Question

Question

This question was previously asked in

NDA-I (Mathematics) Official Paper (Held On: 13 Apr, 2025)

Answer 1

Calculation:

Statement I

$ |M^2| = |M \times M| = |M| \cdot |M| = |M|^2 $

⇒ Statement I is correct.

Statement II

For a non-singular matrix, $ M \times M^{-1} = I $, where $ I $ is the identity matrix.

$ |M| \cdot |M^{-1}| = |I| = 1 $

⇒ Statement II is incorrect unless $ |M| = \pm 1 $.

Statement III

The determinant of a matrix is equal to the determinant of its transpose:

$ |M| = |M^T| $

⇒ Statement III is correct.

Out of the three statements, two are correct: I and III.

Hence, the correct answer is Option 3.

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