When we want to find the inverse of the matrix
$$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$
we're searching for a matrix
$$\begin{bmatrix}x & y \\ z & w\end{bmatrix}$$
such that
$$\begin{bmatrix}x & y \\ z & w\end{bmatrix}\begin{bmatrix}a & b \\ c & d\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\implies$$
$$ax + cy = 1 \\bx + dy = 0 \\az + cw = 0\\bz + dw = 1$$
Finding the inverse is the same as finding the solution for the $x,y$ system and $w,z$ system, which by the cramer's rule only have an unique solution iff
$$det \left(\begin{bmatrix}a & c \\ b & d\end{bmatrix}\right) \neq 0$$
which is the same as saying $$det \left(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\right) \neq 0$$
So here we have the famous result that a matrix is invertible if its determinant is not $0$. But by cramer's rule we know that if the determinant is $0$, we can be in the situation where we could have multiple solutions for $x,y$ and $z,w$ which would imply that a matrix with determinant $0$ could have multiple inverses. So,shouldn't it be false that a matrix is invertible if and only if its determinant is not $0$?