Find the general solution for the determinant of a 3x3 martix. When does the inverse of this matrix not exist?

Let M be a 3x3 matrix s.t. M= |a b c| |g h i| |d e f|

Then Det(M)= a(Det(e,f,h,i))-b(Det(d,f,g,i))+c(Det(d,e,g,h).

Given that the determinant of a 2x2 matrix such as (e,f,h,i) is = ei-fh. The solution is; Det(M)=a(ei-fh)-b(di-fg)+c(dh-eg).

Since the inverse of a matrix, M^-1 = 1/Det(M) * Adj(M), the inverse does not exist when Det(M)=0.