Let X be a square matrix. Consider the following two statements on X: I. X is invertible II. Determinant of X is non-zero Which one of the following is TRUE? A. I implies II; II does not imply I B. II implies I; I does not imply II C. I does not imply II; II does not imply I D. I and II are equivalent statements

GATE 2019 · Engineering Mathematics · Determinant · medium

Answer: D. I and II are equivalent statements

1. Show I implies II (invertible => det != 0): If X is invertible, then X * X^{-1} = I, so det(X) * det(X^{-1}) = 1. Since the product is 1 != 0, det(X) != 0.
2. Show II implies I (det != 0 => invertible): If det(X) != 0, then X^{-1} = (1/det(X)) * adj(X) exists and is well-defined. So X is invertible.
3. Conclusion: Both implications hold: I => II and II => I. Therefore Statements I and II are logically equivalent.