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?

Step-by-step solution

For a non-singular matrix M , |M| 0 . Using the property of determinants |AB| = |A||B| , we have |M^2| = |M M| = |M||M| = |M|^2 . Thus, Statement I is correct. Using the property M M⁻¹ = I , taking the determinant on both sides gives |M M⁻¹| = |I| |M||M⁻¹| = 1 |M⁻¹| = 1 |M| . Since |M| is not necessarily equal to 1 |M| , Statement II is incorrect. The determinant of a matrix is always equal to the determinant of its transpose, so |M| = |M^T| . Thus, Statement III is correct. Exactly two of the given statements are