GATE CSE 2019 | Question: 9

Question

Let $X$ be a square matrix. Consider the following two statements on $X$.

1. $X$ is invertible
2. 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

Comments

What if matrix is not square matrix

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@val_pro20

The inverse of the non-square matrix doesn't exist.

Answer 1

$A^{-1} = \frac{adj A}{\left | A \right |}$

If $\left | A \right |$ = 0; $A^{-1}$ is not defined.

=> If A is invertible, it must have non-zero determinant.

 

If $\left | A \right |$ $\neq 0$ then $A^{-1}$ is defined.

=> If det(A) is non-zero, A must have an inverse.

 

Option D

Answer 2

Inverse of a square matrix only exist if its determinant value is non zero.

Answer 3

Option (D) is Correct .