$X^{-1}=\frac{Adj(X)}{|X|}$

$I)$ 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