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?

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Correct answer: C. Two

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 correct. Answer: Two

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