Check that the matrix is square
Verify that the determinant is nonzero
Use the formula (A^{-1} = frac{1}{det(A)} operatorname{adj}(A)) for small matrices
For a (2 times 2) matrix (begin{bmatrix} a & b \ c & d end{bmatrix}), use (frac{1}{ad-bc}begin{bmatrix} d & -b \ -c & a end{bmatrix})
Use row reduction on ([A mid I]) to transform (A) into the identity matrix
Perform the same row operations on the identity matrix to obtain (A^{-1})
Use Gaussian elimination or Gauss-Jordan elimination for larger matrices
Use a calculator or software for complex matrices
Confirm the result by multiplying (A cdot A^{-1} = I)