GATE CS 2019 Q.19 (CS Q.9): Linear Algebra

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?

I implies II; II does not imply I.

II implies I; I does not imply II.

I does not imply II; II does not imply I.

I and II are equivalent statements.

Official answer and solution

Solution

Recall the fundamental theorem of matrix invertibility connecting non-singularity and non-zero determinant.

For any square matrix $X \in \mathbb{R}^{n \times n}$ , a well-known fundamental property of linear algebra states that:

If $X$ is invertible, then $X X^{-1} = I \implies \det(X) \det(X^{-1}) = \det(I) = 1 \implies \det(X) \ne 0$ .
Conversely, if $\det(X) \ne 0$ , the inverse exists and can be explicitly written as $X^{-1} = \frac{1}{\det(X)} \operatorname{adj}(X)$ .

Therefore, " $X$ is invertible" and "Determinant of $X$ is non-zero" are logically equivalent statements ( $I \iff II$ ).

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.