Let A by an nxn non-singular matrix, then A-1 is the inverse of A. Now (A-1 )-1 =A So the answer is yes.
To find the original matrix of an inverted matrix, simply invert it again. Consider A^-1^-1 = A^1 = A
If an identity matrix is the answer to a problem under matrix multiplication, then each of the two matrices is an inverse matrix of the other.
From Wolfram MathWorld: The inverse of a square matrix A, sometimes called a reciprocal matrix, is a matrix A-1 such that AA-1=I where I is the identity matrix.
The fact that the matrix does not have an inverse does not necessarily mean that none of the variables can be found.
No. A square matrix has an inverse if and only if its determinant is nonzero.