Check that the matrix is square
Compute the determinant
If the determinant is zero, the matrix has no inverse
If the determinant is nonzero, proceed
Use the formula (A^{-1} = frac{1}{det(A)} operatorname{adj}(A))
Find the matrix of cofactors
Transpose the cofactor matrix to get the adjugate
Divide the adjugate by the determinant
For a (2 times 2) matrix (begin{pmatrix} a & b \ c & d end{pmatrix}), use (A^{-1} = frac{1}{ad-bc}begin{pmatrix} d & -b \ -c & a end{pmatrix})
For larger matrices, use row reduction on ([A mid I]) until it becomes ([I mid A^{-1}])
Verify the result by checking that (AA^{-1} = I) and (A^{-1}A = I)