Whenever I needed to find the inverse of a matrix, I was told to check if its determinant is not zero. However, once I directly applied the Gauss-Jordan's method for finding the inverse of matrix whose determinant was zero. The inverse matrix that I got looked pretty normal like any other (if there wasn't a mistake).
I want to know how does the determinant of the matrix is related to inverse of matrix or why is that if determinant is zero then inverse doesn't exist? What exactly is inverse?
Best Answer
It holds that $\det(AB)=\det(A)\det(B)$, so that $\det(A)\det(A^{-1})=1$. In other words, an invertible matrix has (multiplicatively) invertible determinant. (If you work over a field, this means just that the determinant is non-zero.)
On the other hand, if the determinant is invertible, then so is the matrix itself because of the relation to its adjugate.