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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?

1. I implies II; II does not imply I
2. II implies I; I does not imply II
3. I does not imply II; II does not imply I
4. I and II are equivalent statements
edited | 1.6k views

Square Matrix is invertible iff it is non-singular.
So both statements are same.
selected
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$X^{-1}=\frac{Adj(X)}{|X|}$

$I)$ If $X^{-1}$ exist then $|X|\neq0$.

$X^{-1}\implies|X|\neq0$

 $X^{-1}$ $|X|\neq0$. $X^{-1}\implies|X|\neq0$ T T T T F F F T T F F T

$II)$ If $|X|\neq 0$ then $X^{-1}$ exist.

$|X|\neq 0\implies X^{-1}$

 $|X|\neq 0$ $X^{-1}$ $|X|\neq 0\implies X^{-1}$ T T T T F F F T T F F T

we can write like this  $X^{-1}$ exist if only if (or) iff $|X|\neq0$

 $X^{-1}$ $|X|\neq0$ $|X|\neq 0\Longleftrightarrow X^{-1}$ T T T T F F F T F F F T

$I)$ and $II)$ Both are equivalent

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If the option I implies II and II implies I given then I think it would be much suitable
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Yes, you think in a simple way, but IIT professor think some other way

actually what you write is correct but what they ask is also correct.

$P\Longleftrightarrow Q\equiv (P\Longrightarrow Q)\wedge (Q\Longrightarrow P)$
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Thanks
if a square matrix is invertible then it's determinant is non zero

and vice versa

so, (D) is correct option
+1 vote
Option D is right. As Inverse(A) = Adj(A) / Mod(A)

Therefore, if Mod(A) is 0, the Inverse of a matrix cannot be calculated. Therefore both statements are equivalent to each other

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