I can understand the adjugate matrix and the motivation of that to find the inverse, but I can't see how this idea was invented by mathematicians. It's just brilliance or someone understand how the properties were conceived.

I trying to understand this property origins, but I can't explain how cofactor were created too:

$$\sum_{i = 1}^n a_{ij} C_{ik}= \begin{cases} \det(A) & \text{if } j = k \\ 0 & \text{if } j \neq k \end{cases}$$