When doing linear transformation , we see that sometimes we are stretching space and sometimes we are squishing space . Determinant is a measurement of this change .
When det(A) ≠ 0 , after transformation the dimension is still the same .
But when det(A) = 0 , after transformation it is reduced to lower dimension .
Grant from 3b1b does the best job explaining this visually . So , no point of writing more
https://youtu.be/Ip3X9LOh2dk?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab
Take any row / column and for each element add the cofactors . The cofactor is basically the determinant of the matrix excluding the current row and column of matrix multiplied by $-1^{r+c}$
