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