An inverse matrix is any matrix that has the following properties:
Let be a matrix. If there exists a matrix such that:
Then the matrix is invertible and the matrix is an inverse of .
If the matrix is then we can use the following equation to find the inverse:
Notice that the denominator of the fraction above is the determinant of matrix . Source.
Theorem 1: If or than hence is invertible with inverse .
Theorem 2: If is invertible than its inverse is unique.
Theorem 3: If is invertible then for any in the vector is the unique solution to the linear system .