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

For a non-singular matrix $M$, $|M| \neq 0$.

Using the property of determinants $|AB| = |A||B|$, we have $|M^2| = |M \cdot M| = |M||M| = |M|^2$. Thus, Statement I is correct.

Using the property $M M^{-1} = I$, taking the determinant on both sides gives $|M M^{-1}| = |I| \Rightarrow |M||M^{-1}| = 1 \Rightarrow |M^{-1}| = \dfrac{1}{|M|}$. Since $|M|$ is not necessarily equal to $\dfrac{1}{|M|}$, 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 correct.

Answer: Two