linear algebra - Intuition behind Matrix being invertible iff determinant is non-zero

I have been wondering about this question since I was in school. How can one number tell so much about the whole matrix being invertible or not?

I know the proof of this statement now. But I would like to know the intuition behind this result and why this result is actually true.

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The proof I have in mind is:

If $A$ is invertible, then

$$1 = \det(I) = \det(AA^{-1}) = \det(A)\cdot\det(A^{-1})$$

whence $\det(A) \neq 0$.

Conversely, if $\det(A) \neq 0$, we have

$$A adj(A) = adj(A)A = \det(A)I$$

whence $A$ is invertible.

$adj(A)$ is the adjugate matrix of $A$.

$$adj(A)_{ji} = (-1)^{i+j}\det(A_{ij})$$

where $A_{ij}$ is the matrix obtained from $A$ by deleting $ith$ row and $jth$ column.

Any other insightful proofs are also welcome.